Partition function

Symbol: PartitionsP —

p\!\left(n\right)

— Integer partition function

p\!\left(n\right)

denotes the number of ways the integer

n

can be written as a sum of positive integers.

Domain	Codomain
$n \in \mathbb{Z}$	$p\!\left(n\right) \in \mathbb{Z}_{\ge 0}$
$n \in \mathbb{Z}_{\ge 0}$	$p\!\left(n\right) \in \mathbb{Z}_{\ge 1}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("f5e153"),
    SymbolDefinition(PartitionsP, PartitionsP(n), "Integer partition function"),
    Description(PartitionsP(n), "denotes the number of ways the integer", n, "can be written as a sum of positive integers."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(Element(n, ZZ), Element(PartitionsP(n), ZZGreaterEqual(0))), Tuple(Element(n, ZZGreaterEqual(0)), Element(PartitionsP(n), ZZGreaterEqual(1))))))

856db2

Table of

p\!\left(n\right)

for

0 \le n \le 200

$n$	$p\!\left(n\right)$
0	1
1	1
2	2
3	3
4	5
5	7
6	11
7	15
8	22
9	30
10	42
11	56
12	77
13	101
14	135
15	176
16	231
17	297
18	385
19	490
20	627
21	792
22	1002
23	1255
24	1575
25	1958
26	2436
27	3010
28	3718
29	4565
30	5604
31	6842
32	8349
33	10143
34	12310
35	14883
36	17977
37	21637
38	26015
39	31185
40	37338
41	44583
42	53174
43	63261
44	75175
45	89134
46	105558
47	124754
48	147273
49	173525

$n$	$p\!\left(n\right)$
50	204226
51	239943
52	281589
53	329931
54	386155
55	451276
56	526823
57	614154
58	715220
59	831820
60	966467
61	1121505
62	1300156
63	1505499
64	1741630
65	2012558
66	2323520
67	2679689
68	3087735
69	3554345
70	4087968
71	4697205
72	5392783
73	6185689
74	7089500
75	8118264
76	9289091
77	10619863
78	12132164
79	13848650
80	15796476
81	18004327
82	20506255
83	23338469
84	26543660
85	30167357
86	34262962
87	38887673
88	44108109
89	49995925
90	56634173
91	64112359
92	72533807
93	82010177
94	92669720
95	104651419
96	118114304
97	133230930
98	150198136
99	169229875

$n$	$p\!\left(n\right)$
100	190569292
101	214481126
102	241265379
103	271248950
104	304801365
105	342325709
106	384276336
107	431149389
108	483502844
109	541946240
110	607163746
111	679903203
112	761002156
113	851376628
114	952050665
115	1064144451
116	1188908248
117	1327710076
118	1482074143
119	1653668665
120	1844349560
121	2056148051
122	2291320912
123	2552338241
124	2841940500
125	3163127352
126	3519222692
127	3913864295
128	4351078600
129	4835271870
130	5371315400
131	5964539504
132	6620830889
133	7346629512
134	8149040695
135	9035836076
136	10015581680
137	11097645016
138	12292341831
139	13610949895
140	15065878135
141	16670689208
142	18440293320
143	20390982757
144	22540654445
145	24908858009
146	27517052599
147	30388671978
148	33549419497
149	37027355200

$n$	$p\!\left(n\right)$
150	40853235313
151	45060624582
152	49686288421
153	54770336324
154	60356673280
155	66493182097
156	73232243759
157	80630964769
158	88751778802
159	97662728555
160	107438159466
161	118159068427
162	129913904637
163	142798995930
164	156919475295
165	172389800255
166	189334822579
167	207890420102
168	228204732751
169	250438925115
170	274768617130
171	301384802048
172	330495499613
173	362326859895
174	397125074750
175	435157697830
176	476715857290
177	522115831195
178	571701605655
179	625846753120
180	684957390936
181	749474411781
182	819876908323
183	896684817527
184	980462880430
185	1071823774337
186	1171432692373
187	1280011042268
188	1398341745571
189	1527273599625
190	1667727404093
191	1820701100652
192	1987276856363
193	2168627105469
194	2366022741845
195	2580840212973
196	2814570987591
197	3068829878530
198	3345365983698
199	3646072432125
200	3972999029388

Table data:

\left(n, y\right)

such that

p\!\left(n\right) = y

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function

Source code for this entry:

Entry(ID("856db2"),
    Description("Table of", PartitionsP(n), "for", LessEqual(0, n, 200)),
    Table(TableRelation(Tuple(n, y), Equal(PartitionsP(n), y)), TableHeadings(n, PartitionsP(n)), TableSplit(4), List(Tuple(0, 1), Tuple(1, 1), Tuple(2, 2), Tuple(3, 3), Tuple(4, 5), Tuple(5, 7), Tuple(6, 11), Tuple(7, 15), Tuple(8, 22), Tuple(9, 30), Tuple(10, 42), Tuple(11, 56), Tuple(12, 77), Tuple(13, 101), Tuple(14, 135), Tuple(15, 176), Tuple(16, 231), Tuple(17, 297), Tuple(18, 385), Tuple(19, 490), Tuple(20, 627), Tuple(21, 792), Tuple(22, 1002), Tuple(23, 1255), Tuple(24, 1575), Tuple(25, 1958), Tuple(26, 2436), Tuple(27, 3010), Tuple(28, 3718), Tuple(29, 4565), Tuple(30, 5604), Tuple(31, 6842), Tuple(32, 8349), Tuple(33, 10143), Tuple(34, 12310), Tuple(35, 14883), Tuple(36, 17977), Tuple(37, 21637), Tuple(38, 26015), Tuple(39, 31185), Tuple(40, 37338), Tuple(41, 44583), Tuple(42, 53174), Tuple(43, 63261), Tuple(44, 75175), Tuple(45, 89134), Tuple(46, 105558), Tuple(47, 124754), Tuple(48, 147273), Tuple(49, 173525), Tuple(50, 204226), Tuple(51, 239943), Tuple(52, 281589), Tuple(53, 329931), Tuple(54, 386155), Tuple(55, 451276), Tuple(56, 526823), Tuple(57, 614154), Tuple(58, 715220), Tuple(59, 831820), Tuple(60, 966467), Tuple(61, 1121505), Tuple(62, 1300156), Tuple(63, 1505499), Tuple(64, 1741630), Tuple(65, 2012558), Tuple(66, 2323520), Tuple(67, 2679689), Tuple(68, 3087735), Tuple(69, 3554345), Tuple(70, 4087968), Tuple(71, 4697205), Tuple(72, 5392783), Tuple(73, 6185689), Tuple(74, 7089500), Tuple(75, 8118264), Tuple(76, 9289091), Tuple(77, 10619863), Tuple(78, 12132164), Tuple(79, 13848650), Tuple(80, 15796476), Tuple(81, 18004327), Tuple(82, 20506255), Tuple(83, 23338469), Tuple(84, 26543660), Tuple(85, 30167357), Tuple(86, 34262962), Tuple(87, 38887673), Tuple(88, 44108109), Tuple(89, 49995925), Tuple(90, 56634173), Tuple(91, 64112359), Tuple(92, 72533807), Tuple(93, 82010177), Tuple(94, 92669720), Tuple(95, 104651419), Tuple(96, 118114304), Tuple(97, 133230930), Tuple(98, 150198136), Tuple(99, 169229875), Tuple(100, 190569292), Tuple(101, 214481126), Tuple(102, 241265379), Tuple(103, 271248950), Tuple(104, 304801365), Tuple(105, 342325709), Tuple(106, 384276336), Tuple(107, 431149389), Tuple(108, 483502844), Tuple(109, 541946240), Tuple(110, 607163746), Tuple(111, 679903203), Tuple(112, 761002156), Tuple(113, 851376628), Tuple(114, 952050665), Tuple(115, 1064144451), Tuple(116, 1188908248), Tuple(117, 1327710076), Tuple(118, 1482074143), Tuple(119, 1653668665), Tuple(120, 1844349560), Tuple(121, 2056148051), Tuple(122, 2291320912), Tuple(123, 2552338241), Tuple(124, 2841940500), Tuple(125, 3163127352), Tuple(126, 3519222692), Tuple(127, 3913864295), Tuple(128, 4351078600), Tuple(129, 4835271870), Tuple(130, 5371315400), Tuple(131, 5964539504), Tuple(132, 6620830889), Tuple(133, 7346629512), Tuple(134, 8149040695), Tuple(135, 9035836076), Tuple(136, 10015581680), Tuple(137, 11097645016), Tuple(138, 12292341831), Tuple(139, 13610949895), Tuple(140, 15065878135), Tuple(141, 16670689208), Tuple(142, 18440293320), Tuple(143, 20390982757), Tuple(144, 22540654445), Tuple(145, 24908858009), Tuple(146, 27517052599), Tuple(147, 30388671978), Tuple(148, 33549419497), Tuple(149, 37027355200), Tuple(150, 40853235313), Tuple(151, 45060624582), Tuple(152, 49686288421), Tuple(153, 54770336324), Tuple(154, 60356673280), Tuple(155, 66493182097), Tuple(156, 73232243759), Tuple(157, 80630964769), Tuple(158, 88751778802), Tuple(159, 97662728555), Tuple(160, 107438159466), Tuple(161, 118159068427), Tuple(162, 129913904637), Tuple(163, 142798995930), Tuple(164, 156919475295), Tuple(165, 172389800255), Tuple(166, 189334822579), Tuple(167, 207890420102), Tuple(168, 228204732751), Tuple(169, 250438925115), Tuple(170, 274768617130), Tuple(171, 301384802048), Tuple(172, 330495499613), Tuple(173, 362326859895), Tuple(174, 397125074750), Tuple(175, 435157697830), Tuple(176, 476715857290), Tuple(177, 522115831195), Tuple(178, 571701605655), Tuple(179, 625846753120), Tuple(180, 684957390936), Tuple(181, 749474411781), Tuple(182, 819876908323), Tuple(183, 896684817527), Tuple(184, 980462880430), Tuple(185, 1071823774337), Tuple(186, 1171432692373), Tuple(187, 1280011042268), Tuple(188, 1398341745571), Tuple(189, 1527273599625), Tuple(190, 1667727404093), Tuple(191, 1820701100652), Tuple(192, 1987276856363), Tuple(193, 2168627105469), Tuple(194, 2366022741845), Tuple(195, 2580840212973), Tuple(196, 2814570987591), Tuple(197, 3068829878530), Tuple(198, 3345365983698), Tuple(199, 3646072432125), Tuple(200, 3972999029388))))

cebe1b

p\!\left(0\right) = \left|\left\{\left[\right]\right\}\right| = 1

TeX:

p\!\left(0\right) = \left|\left\{\left[\right]\right\}\right| = 1

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Cardinality`	$\left\|S\right\|$	Set cardinality

Source code for this entry:

Entry(ID("cebe1b"),
    Formula(Equal(PartitionsP(0), Cardinality(Set(List())), 1)))

e84642

p\!\left(1\right) = \left|\left\{\left[1\right]\right\}\right| = 1

TeX:

p\!\left(1\right) = \left|\left\{\left[1\right]\right\}\right| = 1

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Cardinality`	$\left\|S\right\|$	Set cardinality

Source code for this entry:

Entry(ID("e84642"),
    Formula(Equal(PartitionsP(1), Cardinality(Set(List(1))), 1)))

b2583f

p\!\left(2\right) = \left|\left\{\left[2\right], \left[1, 1\right]\right\}\right| = 2

TeX:

p\!\left(2\right) = \left|\left\{\left[2\right], \left[1, 1\right]\right\}\right| = 2

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Cardinality`	$\left\|S\right\|$	Set cardinality

Source code for this entry:

Entry(ID("b2583f"),
    Formula(Equal(PartitionsP(2), Cardinality(Set(List(2), List(1, 1))), 2)))

7ef291

p\!\left(3\right) = \left|\left\{\left[3\right], \left[2, 1\right], \left[1, 1, 1\right]\right\}\right| = 3

TeX:

p\!\left(3\right) = \left|\left\{\left[3\right], \left[2, 1\right], \left[1, 1, 1\right]\right\}\right| = 3

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Cardinality`	$\left\|S\right\|$	Set cardinality

Source code for this entry:

Entry(ID("7ef291"),
    Formula(Equal(PartitionsP(3), Cardinality(Set(List(3), List(2, 1), List(1, 1, 1))), 3)))

6018a4

p\!\left(4\right) = \left|\left\{\left[4\right], \left[3, 1\right], \left[2, 2\right], \left[2, 1, 1\right], \left[1, 1, 1, 1\right]\right\}\right| = 5

TeX:

p\!\left(4\right) = \left|\left\{\left[4\right], \left[3, 1\right], \left[2, 2\right], \left[2, 1, 1\right], \left[1, 1, 1, 1\right]\right\}\right| = 5

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Cardinality`	$\left\|S\right\|$	Set cardinality

Source code for this entry:

Entry(ID("6018a4"),
    Formula(Equal(PartitionsP(4), Cardinality(Set(List(4), List(3, 1), List(2, 2), List(2, 1, 1), List(1, 1, 1, 1))), 5)))

cd3013

p\!\left(-n\right) = 0

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

p\!\left(-n\right) = 0

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("cd3013"),
    Formula(Equal(PartitionsP(Neg(n)), 0)),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

9933df

Table of

p\!\left({10}^{n}\right)

to 50 digits for

0 \le n \le 30

$n$	$p\!\left({10}^{n}\right)$
0	1
1	42
2	190569292
3	24061467864032622473692149727991
4	3.6167251325636293988820471890953695495016030339316 · 10¹⁰⁶
5	2.7493510569775696512677516320986352688173429315980 · 10³⁴⁶
6	1.4716849863582233986310047606098959434840304844391 · 10¹¹⁰⁷
7	9.2027175502604546685596278166825605430729405281024 · 10³⁵¹⁴
8	1.7605170459462491413603738946791352040098537975109 · 10¹¹¹³¹
9	1.6045350842809668832728039026391874671468439447108 · 10³⁵²¹⁸
10	1.0523943461106485297281294178237273482933553642403 · 10¹¹¹³⁹⁰
11	4.1604280503811938572793734321866528100080985902856 · 10³⁵²²⁶⁸
12	6.1290009628366844179973253747618396500221302871150 · 10^1113995
13	5.7144146870758614917950406422638086360770375255550 · 10^3522790
14	2.7509605970815655120620992887934278296645559629575 · 10^11140071
15	1.3655377298964220782966300424326842827176530525453 · 10^35228030
16	9.1291313906814503700935608040674225211147823841734 · 10^111400845
17	8.2913007910135095775713801190603101231989771169282 · 10^352280441
18	1.4787003107715742179708592460012268624667759844895 · 10^1114008609
19	5.6469284039962075996762611156427010823552403269436 · 10^3522804577
20	1.8381765083448826436460575151963949703661288601871 · 10^11140086259
21	1.2125743672403400786494500161173864623062685147724 · 10^35228045954
22	1.6197861609669294695161189248758019106925992523025 · 10^111400862778
23	2.5273733524499047268270064364643395566828146205663 · 10^352280459735
24	4.5725915523567534123265286016382336070839930153380 · 10^{1114008627985}
25	3.9109259209775087194782941921388925892278301362731 · 10^{3522804597566}
26	1.4696356043302577340385578467919062215335762286639 · 10^{11140086280078}
27	3.0787999182688279161294058462619983591578972390067 · 10^{35228045975896}
28	1.7285510783890260357320054674456196730673005602418 · 10^{111400862801021}
29	2.8144933818546523144681227969465158737560857425620 · 10^{352280459759213}
30	8.7580564911459301179252748158578897130776558175089 · 10^{1114008628010469}

Table data:

\left(n, y\right)

such that

\operatorname{NearestDecimal}\!\left(p\!\left({10}^{n}\right), 50\right) = y

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Pow`	${a}^{b}$	Power

Source code for this entry:

Entry(ID("9933df"),
    Description("Table of", PartitionsP(Pow(10, n)), "to 50 digits for", LessEqual(0, n, 30)),
    Table(TableRelation(Tuple(n, y), Equal(NearestDecimal(PartitionsP(Pow(10, n)), 50), y)), TableHeadings(n, PartitionsP(Pow(10, n))), TableSplit(1), List(Tuple(0, Decimal("1")), Tuple(1, Decimal("42")), Tuple(2, Decimal("190569292")), Tuple(3, Decimal("24061467864032622473692149727991")), Tuple(4, Decimal("3.6167251325636293988820471890953695495016030339316e+106")), Tuple(5, Decimal("2.7493510569775696512677516320986352688173429315980e+346")), Tuple(6, Decimal("1.4716849863582233986310047606098959434840304844391e+1107")), Tuple(7, Decimal("9.2027175502604546685596278166825605430729405281024e+3514")), Tuple(8, Decimal("1.7605170459462491413603738946791352040098537975109e+11131")), Tuple(9, Decimal("1.6045350842809668832728039026391874671468439447108e+35218")), Tuple(10, Decimal("1.0523943461106485297281294178237273482933553642403e+111390")), Tuple(11, Decimal("4.1604280503811938572793734321866528100080985902856e+352268")), Tuple(12, Decimal("6.1290009628366844179973253747618396500221302871150e+1113995")), Tuple(13, Decimal("5.7144146870758614917950406422638086360770375255550e+3522790")), Tuple(14, Decimal("2.7509605970815655120620992887934278296645559629575e+11140071")), Tuple(15, Decimal("1.3655377298964220782966300424326842827176530525453e+35228030")), Tuple(16, Decimal("9.1291313906814503700935608040674225211147823841734e+111400845")), Tuple(17, Decimal("8.2913007910135095775713801190603101231989771169282e+352280441")), Tuple(18, Decimal("1.4787003107715742179708592460012268624667759844895e+1114008609")), Tuple(19, Decimal("5.6469284039962075996762611156427010823552403269436e+3522804577")), Tuple(20, Decimal("1.8381765083448826436460575151963949703661288601871e+11140086259")), Tuple(21, Decimal("1.2125743672403400786494500161173864623062685147724e+35228045954")), Tuple(22, Decimal("1.6197861609669294695161189248758019106925992523025e+111400862778")), Tuple(23, Decimal("2.5273733524499047268270064364643395566828146205663e+352280459735")), Tuple(24, Decimal("4.5725915523567534123265286016382336070839930153380e+1114008627985")), Tuple(25, Decimal("3.9109259209775087194782941921388925892278301362731e+3522804597566")), Tuple(26, Decimal("1.4696356043302577340385578467919062215335762286639e+11140086280078")), Tuple(27, Decimal("3.0787999182688279161294058462619983591578972390067e+35228045975896")), Tuple(28, Decimal("1.7285510783890260357320054674456196730673005602418e+111400862801021")), Tuple(29, Decimal("2.8144933818546523144681227969465158737560857425620e+352280459759213")), Tuple(30, Decimal("8.7580564911459301179252748158578897130776558175089e+1114008628010469")))))

599417

\sum_{n=0}^{\infty} p\!\left(n\right) {q}^{n} = \frac{1}{\phi\!\left(q\right)}

Assumptions:

q \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|q\right| \lt 1

TeX:

\sum_{n=0}^{\infty} p\!\left(n\right) {q}^{n} = \frac{1}{\phi\!\left(q\right)}

q \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|q\right| \lt 1

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`EulerQSeries`	$\phi\!\left(q\right)$	Euler's q-series
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("599417"),
    Formula(Equal(Sum(Mul(PartitionsP(n), Pow(q, n)), Tuple(n, 0, Infinity)), Div(1, EulerQSeries(q)))),
    Variables(q),
    Assumptions(And(Element(q, CC), Less(Abs(q), 1))))

acdce8

p\!\left(n\right) = \sum_{k=1}^{n + 1} {\left(-1\right)}^{k + 1} \left(p\!\left(n - \frac{k \left(3 k - 1\right)}{2}\right) + p\!\left(n - \frac{k \left(3 k + 1\right)}{2}\right)\right)

Assumptions:

n \in \mathbb{Z}

TeX:

p\!\left(n\right) = \sum_{k=1}^{n + 1} {\left(-1\right)}^{k + 1} \left(p\!\left(n - \frac{k \left(3 k - 1\right)}{2}\right) + p\!\left(n - \frac{k \left(3 k + 1\right)}{2}\right)\right)

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("acdce8"),
    Formula(Equal(PartitionsP(n), Sum(Mul(Pow(-1, Add(k, 1)), Add(PartitionsP(Sub(n, Div(Mul(k, Sub(Mul(3, k), 1)), 2))), PartitionsP(Sub(n, Div(Mul(k, Add(Mul(3, k), 1)), 2))))), Tuple(k, 1, Add(n, 1))))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

4d2e45

p\!\left(n\right) = \frac{1}{n} \sum_{k=0}^{n - 1} \sigma\!\left(n - k\right) p\!\left(k\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

p\!\left(n\right) = \frac{1}{n} \sum_{k=0}^{n - 1} \sigma\!\left(n - k\right) p\!\left(k\right)

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`DivisorSigma`	$\sigma\!\left(n\right)$	Sum of divisors function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("4d2e45"),
    Formula(Equal(PartitionsP(n), Mul(Div(1, n), Sum(Mul(DivisorSigma(Sub(n, k)), PartitionsP(k)), Tuple(k, 0, Sub(n, 1)))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

d8e37d

p\!\left(5 n + 4\right) \bmod 5 = 0

Assumptions:

n \in \mathbb{Z}

TeX:

p\!\left(5 n + 4\right) \bmod 5 = 0

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d8e37d"),
    Formula(Equal(Mod(PartitionsP(Add(Mul(5, n), 4)), 5), 0)),
    Variables(n),
    Assumptions(Element(n, ZZ)))

89260d

p\!\left(7 n + 5\right) \bmod 7 = 0

Assumptions:

n \in \mathbb{Z}

TeX:

p\!\left(7 n + 5\right) \bmod 7 = 0

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("89260d"),
    Formula(Equal(Mod(PartitionsP(Add(Mul(7, n), 5)), 7), 0)),
    Variables(n),
    Assumptions(Element(n, ZZ)))

dacd74

p\!\left(11 n + 6\right) \bmod 11 = 0

Assumptions:

n \in \mathbb{Z}

TeX:

p\!\left(11 n + 6\right) \bmod 11 = 0

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("dacd74"),
    Formula(Equal(Mod(PartitionsP(Add(Mul(11, n), 6)), 11), 0)),
    Variables(n),
    Assumptions(Element(n, ZZ)))

f7407a

p\!\left(n\right) \le p\!\left(n + 1\right)

Assumptions:

n \in \mathbb{Z}

TeX:

p\!\left(n\right) \le p\!\left(n + 1\right)

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("f7407a"),
    Formula(LessEqual(PartitionsP(n), PartitionsP(Add(n, 1)))),
    Variables(n),
    Assumptions(Element(n, ZZ)))

df3c07

p\!\left(n\right) \lt p\!\left(n + 1\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

p\!\left(n\right) \lt p\!\left(n + 1\right)

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("df3c07"),
    Formula(Less(PartitionsP(n), PartitionsP(Add(n, 1)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

d72123

p\!\left(n\right) \ge n

Assumptions:

n \in \mathbb{Z}

TeX:

p\!\left(n\right) \ge n

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d72123"),
    Formula(GreaterEqual(PartitionsP(n), n)),
    Variables(n),
    Assumptions(Element(n, ZZ)))

e1f15b

p\!\left(n\right) \le {2}^{n}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

p\!\left(n\right) \le {2}^{n}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("e1f15b"),
    Formula(LessEqual(PartitionsP(n), Pow(2, n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

7697af

p\!\left(n\right) \sim \frac{{e}^{\pi \sqrt{2 n / 3}}}{4 n \sqrt{3}}, \; n \to \infty

Assumptions:

n \in \mathbb{Z}

TeX:

p\!\left(n\right) \sim \frac{{e}^{\pi \sqrt{2 n / 3}}}{4 n \sqrt{3}}, \; n \to \infty

n \in \mathbb{Z}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`Sqrt`	$\sqrt{z}$	Principal square root
`Infinity`	$\infty$	Positive infinity
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("7697af"),
    Formula(AsymptoticTo(PartitionsP(n), Div(Exp(Mul(ConstPi, Sqrt(Div(Mul(2, n), 3)))), Mul(Mul(4, n), Sqrt(3))), n, Infinity)),
    Variables(n),
    Assumptions(Element(n, ZZ)))

fb7a63

p\!\left(n\right) = \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{\infty} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

p\!\left(n\right) = \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{\infty} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ConstPi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`HardyRamanujanA`	$A\!\left(n, k\right)$	Exponential sum in the Hardy-Ramanujan-Rademacher formula
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("fb7a63"),
    Formula(Equal(PartitionsP(n), Mul(Div(Mul(2, ConstPi), Pow(Sub(Mul(24, n), 1), Div(3, 4))), Sum(Mul(Div(HardyRamanujanA(n, k), k), BesselI(Div(3, 2), Mul(Div(ConstPi, k), Sqrt(Mul(Div(2, 3), Sub(n, Div(1, 24))))))), Tuple(k, 1, Infinity))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

3eae25

Symbol: HardyRamanujanA —

A\!\left(n, k\right)

— Exponential sum in the Hardy-Ramanujan-Rademacher formula

Domain	Codomain
$n \in \mathbb{Z}_{\ge 0} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 1}$	$A\!\left(n, k\right) \in \mathbb{R}$

Table data:

\left(P, Q\right)

such that

\left(P\right) \implies \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`HardyRamanujanA`	$A\!\left(n, k\right)$	Exponential sum in the Hardy-Ramanujan-Rademacher formula
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("3eae25"),
    SymbolDefinition(HardyRamanujanA, HardyRamanujanA(n, k), "Exponential sum in the Hardy-Ramanujan-Rademacher formula"),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(1))), Element(HardyRamanujanA(n, k), RR)))))

5adbc3

A\!\left(n, k\right) = \sum_{r=0}^{k - 1} \delta_{(\gcd\left(r, k\right),1)} \exp\!\left(\pi i \left(s\!\left(r, k\right) - \frac{2 n r}{k}\right)\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 1}

TeX:

A\!\left(n, k\right) = \sum_{r=0}^{k - 1} \delta_{(\gcd\left(r, k\right),1)} \exp\!\left(\pi i \left(s\!\left(r, k\right) - \frac{2 n r}{k}\right)\right)

n \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`HardyRamanujanA`	$A\!\left(n, k\right)$	Exponential sum in the Hardy-Ramanujan-Rademacher formula
`KroneckerDelta`	$\delta_{(x,y)}$	Kronecker delta
`GCD`	$\gcd\!\left(n, k\right)$	Greatest common divisor
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`DedekindSum`	$s\!\left(n, k\right)$	Dedekind sum
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("5adbc3"),
    Formula(Equal(HardyRamanujanA(n, k), Sum(Mul(KroneckerDelta(GCD(r, k), 1), Exp(Mul(Mul(ConstPi, ConstI), Sub(DedekindSum(r, k), Div(Mul(Mul(2, n), r), k))))), Tuple(r, 0, Sub(k, 1))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(1)), Element(k, ZZGreaterEqual(1)))))

afd27a

\left|p\!\left(n\right) - \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{N} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)\right| \le \frac{44 {\pi}^{2}}{225 \sqrt{3 N}} + \frac{\pi \sqrt{2}}{75} \sqrt{\frac{N}{n - 1}} \sinh\!\left(\frac{\pi}{N} \sqrt{\frac{2 n}{3}}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 2} \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 1}

TeX:

\left|p\!\left(n\right) - \frac{2 \pi}{{\left(24 n - 1\right)}^{3 / 4}} \sum_{k=1}^{N} \frac{A\!\left(n, k\right)}{k} I_{3 / 2}\!\left(\frac{\pi}{k} \sqrt{\frac{2}{3} \left(n - \frac{1}{24}\right)}\right)\right| \le \frac{44 {\pi}^{2}}{225 \sqrt{3 N}} + \frac{\pi \sqrt{2}}{75} \sqrt{\frac{N}{n - 1}} \sinh\!\left(\frac{\pi}{N} \sqrt{\frac{2 n}{3}}\right)

n \in \mathbb{Z}_{\ge 2} \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`PartitionsP`	$p\!\left(n\right)$	Integer partition function
`ConstPi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`HardyRamanujanA`	$A\!\left(n, k\right)$	Exponential sum in the Hardy-Ramanujan-Rademacher formula
`BesselI`	$I_{\nu}\!\left(z\right)$	Modified Bessel function of the first kind
`Sqrt`	$\sqrt{z}$	Principal square root
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("afd27a"),
    Formula(LessEqual(Abs(Sub(PartitionsP(n), Mul(Div(Mul(2, ConstPi), Pow(Sub(Mul(24, n), 1), Div(3, 4))), Sum(Mul(Div(HardyRamanujanA(n, k), k), BesselI(Div(3, 2), Mul(Div(ConstPi, k), Sqrt(Mul(Div(2, 3), Sub(n, Div(1, 24))))))), Tuple(k, 1, N))))), Add(Div(Mul(44, Pow(ConstPi, 2)), Mul(225, Sqrt(Mul(3, N)))), Mul(Mul(Div(Mul(ConstPi, Sqrt(2)), 75), Sqrt(Div(N, Sub(n, 1)))), Sinh(Mul(Div(ConstPi, N), Sqrt(Div(Mul(2, n), 3)))))))),
    Variables(n, N),
    Assumptions(And(Element(n, ZZGreaterEqual(2)), Element(N, ZZGreaterEqual(1)))))

Specific values

Generating functions

Sums and recurrence relations

Congruences

Inequalities

Asymptotic expansions

Hardy-Ramanujan-Rademacher formula