Integer sequences

Symbol: SloaneA —

\text{A00000X}\!\left(n\right)

— Sequence X in Sloane's OEIS

SloaneA(X, n), rendered as

\text{A00000X}\!\left(n\right)

gives the integer at position

n

in sequence number

X

in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS). The identifier

X

can be specified as an integer (e.g. 55) or a text (e.g. "A000055")

Semantically, this function represents the intended infinite extension of each (non-finite) OEIS sequence although the OEIS database itself of course only lists a finite number of terms.

Definitions:

Fungrim symbol	Notation	Short description
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS

Source code for this entry:

Entry(ID("aac67f"),
    SymbolDefinition(SloaneA, SloaneA(X, n), "Sequence X in Sloane's OEIS"),
    Description(SourceForm(SloaneA(X, n)), ", rendered as", SloaneA(X, n), "gives the integer at position", n, "in sequence number", X, "in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS). ", "The identifier", X, "can be specified as an integer (e.g. 55) or a text (e.g. \"A000055\")"),
    Description("Semantically, this function represents the intended infinite extension of each (non-finite) ", "OEIS sequence although the OEIS database itself of course only lists a finite number of terms."))

963387

\left(X \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}\right) \;\implies\; \text{A00000X}\!\left(n\right) \in \mathbb{Z} \cup \left\{\operatorname{Undefined}\right\}

TeX:

\left(X \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}\right) \;\implies\; \text{A00000X}\!\left(n\right) \in \mathbb{Z} \cup \left\{\operatorname{Undefined}\right\}

Definitions:

Fungrim symbol	Notation	Short description
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS

Source code for this entry:

Entry(ID("963387"),
    Formula(Implies(And(Element(X, ZZGreaterEqual(1)), Element(n, ZZ)), Element(SloaneA(X, n), Union(ZZ, Set(Undefined))))),
    Variables(X, n))

9d0839

p_{n} = \text{A000040}\!\left(n\right)

Assumptions:

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

References:

Sequence A000040 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

p_{n} = \text{A000040}\!\left(n\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`PrimeNumber`	$p_{n}$	nth prime number
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("9d0839"),
    Formula(Equal(PrimeNumber(n), SloaneA("A000040", n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

8eed2c

p(n) = \text{A000041}\!\left(n\right)

Assumptions:

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

References:

Sequence A000041 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

p(n) = \text{A000041}\!\left(n\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`PartitionsP`	$p(n)$	Integer partition function
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("8eed2c"),
    Formula(Equal(PartitionsP(n), SloaneA("A000041", n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

373aa1

F_{n} = \text{A000045}\!\left(n\right)

Assumptions:

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

References:

Sequence A000045 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

F_{n} = \text{A000045}\!\left(n\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`Fibonacci`	$F_{n}$	Fibonacci number
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("373aa1"),
    Formula(Equal(Fibonacci(n), SloaneA("A000045", n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

60dc3e

B_{n} = \text{A000110}\!\left(n\right)

Assumptions:

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

References:

Sequence A000110 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

B_{n} = \text{A000110}\!\left(n\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`BellNumber`	$B_{n}$	Bell number
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("60dc3e"),
    Formula(Equal(BellNumber(n), SloaneA("A000110", n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

d12aa0

n ! = \text{A000142}\!\left(n\right)

Assumptions:

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

References:

Sequence A000142 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

n ! = \text{A000142}\!\left(n\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`Factorial`	$n !$	Factorial
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("d12aa0"),
    Formula(Equal(Factorial(n), SloaneA("A000142", n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

4fa169

\pi(n) = \text{A000720}\!\left(n\right)

Assumptions:

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

References:

Sequence A000720 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

\pi(n) = \text{A000720}\!\left(n\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`PrimePi`	$\pi(x)$	Prime counting function
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("4fa169"),
    Formula(Equal(PrimePi(n), SloaneA("A000720", n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

6af603

g(n) = \text{A000793}\!\left(n\right)

Assumptions:

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

References:

Sequence A000793 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

g(n) = \text{A000793}\!\left(n\right)

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

Definitions:

Fungrim symbol	Notation	Short description
`LandauG`	$g(n)$	Landau's function
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("6af603"),
    Formula(Equal(LandauG(n), SloaneA("A000793", n))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

483547

\pi = \sum_{n=1}^{\infty} \text{A000796}\!\left(n\right) {10}^{1 - n}

References:

Sequence A000796 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

\pi = \sum_{n=1}^{\infty} \text{A000796}\!\left(n\right) {10}^{1 - n}

Definitions:

Fungrim symbol	Notation	Short description
`Pi`	$\pi$	The constant pi (3.14...)
`Sum`	$\sum_{n} f(n)$	Sum
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity

Source code for this entry:

Entry(ID("483547"),
    Formula(Equal(Pi, Sum(Mul(SloaneA("A000796", n), Pow(10, Sub(1, n))), For(n, 1, Infinity)))))

b6111c

B_{n} = \frac{\text{A027641}\!\left(n\right)}{\text{A027642}\!\left(n\right)}

Assumptions:

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

References:

Sequence A027641 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)
Sequence A027642 in Sloane's On-Line Encyclopedia of Integer Sequences (OEIS)

TeX:

B_{n} = \frac{\text{A027641}\!\left(n\right)}{\text{A027642}\!\left(n\right)}

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

Definitions:

Fungrim symbol	Notation	Short description
`BernoulliB`	$B_{n}$	Bernoulli number
`SloaneA`	$\text{A00000X}\!\left(n\right)$	Sequence X in Sloane's OEIS
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("b6111c"),
    Formula(Equal(BernoulliB(n), Div(SloaneA("A027641", n), SloaneA("A027642", n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

Definitions

Core sequences