chunky v0.9.0 Chunky.Sequence.OEIS.Core View Source

OEIS Core Sequences.

Available Sequences

Link to this section Summary

Functions

OEIS Sequence A000005 - Number of divisors of N, simga-0(n), 𝝈0(n).

OEIS Sequence A000009 - Number of partitions of n into distinct parts

OEIS Sequence A000010 - Euler's totient function phi(n)

OEIS Sequence A000041 - Partitions of integer N

OEIS Sequence A000079 - Powers of 2 a(n) = 2^n

OEIS Sequence A000203 - Sum of Divisors σ1(n)

OEIS Sequence A000396 - Perfect Numbers

OEIS Sequence A000593 - Sum of Odd Divisors of N

OEIS Sequence A001065 - Sum of proper divisors (Aliquot parts) of N.

OEIS Sequence A001157 - Sum of squares of divisors of N, simga-2(n), 𝝈2(n).

OEIS Sequence A005100 - Deficient Numbers

OEIS Sequence A005101 - Abundant Numbers

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create_sequence_a000005(opts)

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OEIS Sequence A000005 - Number of divisors of N, simga-0(n), 𝝈0(n).

From OEIS A000005:

d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n. (Formerly M0246 N0086)

Sequence IDs: :a000005

Finite: False

Offset: 1

Example

iex> Sequence.create(Sequence.OEIS.Core, :a000005) |> Sequence.take!(10)
[1, 2, 2, 3, 2, 4, 2, 4, 3, 4]
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create_sequence_a000009(opts)

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OEIS Sequence A000009 - Number of partitions of n into distinct parts

From OEIS A000009:

Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts (if n > 0). (Formerly M0281 N0100)

Divergence

Calculation of this sequence is based on translation of a Maxima program by Vladimir Kruchinin, and diverges from canonical results for n > 10.

Sequence IDs: :a000009

Finite: False

Offset: 0

Example

iex> Sequence.create(Sequence.OEIS.Core, :a000009) |> Sequence.take!(10)
[1, 1, 1, 2, 2, 3, 4, 5, 6, 8]
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create_sequence_a000010(opts)

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OEIS Sequence A000010 - Euler's totient function phi(n)

From OEIS A000010:

Euler totient function phi(n): count numbers <= n and prime to n. (Formerly M0299 N0111)

Sequence IDs: :a000010

Finite: false

Example

iex> Sequence.create(Sequence.OEIS.Core, :a000010) |> Sequence.take!(20)
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8]
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create_sequence_a000041(opts)

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OEIS Sequence A000041 - Partitions of integer N

This sequence contains the partitions of the integers from 0 to 250.

From Wikipedia:

In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions: 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4.

From OEIS A000041:

a(n) is the number of partitions of n (the partition numbers). (Formerly M0663 N0244)

Sequence IDs: :a000041

Finite: true

Example

iex> Sequence.create(Sequence.OEIS.Core, :a000041) |> Sequence.take!(10)
[1, 1, 2, 3, 5, 7, 11, 15, 22, 30]
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create_sequence_a000079(opts)

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OEIS Sequence A000079 - Powers of 2 a(n) = 2^n

From OEIS A000009:

Powers of 2: a(n) = 2^n. (Formerly M1129 N0432)

Sequence IDs: :a000079

Finite: False

Offset: 0

Example

iex> Sequence.create(Sequence.OEIS.Core, :a000079) |> Sequence.take!(20)
[1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288]
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create_sequence_a000203(opts)

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OEIS Sequence A000203 - Sum of Divisors σ1(n)

From OEIS A000203:

(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n). (Formerly M2329 N0921)

Sequence IDs: :a000203

Finite: False

Offset: 1

Example

iex> Sequence.create(Sequence.OEIS.Core, :a000203) |> Sequence.take!(20)
[1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42]  
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create_sequence_a000396(opts)

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OEIS Sequence A000396 - Perfect Numbers

From OEIS A000396:

Perfect numbers n: n is equal to the sum of the proper divisors of n. (Formerly M4186 N1744)

Sequence IDs: :a000396

Finite: True

Offset: 1

Example

iex> Sequence.create(Sequence.OEIS.Core, :a000396) |> Sequence.take!(5)
[6, 28, 496, 8128, 33550336]  
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create_sequence_a000593(opts)

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OEIS Sequence A000593 - Sum of Odd Divisors of N

From OEIS A000593:

Sum of odd divisors of n. (Formerly M3197 N1292)

Sequence IDs: :a000593

Finite: False

Offset: 1

Example

iex> Sequence.create(Sequence.OEIS.Core, :a000593) |> Sequence.take!(10)
[1, 1, 4, 1, 6, 4, 8, 1, 13, 6]
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create_sequence_a001065(opts)

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OEIS Sequence A001065 - Sum of proper divisors (Aliquot parts) of N.

From OEIS A001065:

Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n. (Formerly M2226 N0884)

Sequence IDs: :a001065

Finite: False

Offset: 1

Example

iex> Sequence.create(Sequence.OEIS.Core, :a001065) |> Sequence.take!(20)
[0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22]
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create_sequence_a001157(opts)

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OEIS Sequence A001157 - Sum of squares of divisors of N, simga-2(n), 𝝈2(n).

From OEIS A001157:

sigma_2(n): sum of squares of divisors of n. (Formerly M3799 N1551)

Sequence IDs: :a001157

Finite: False

Offset: 1

Example

iex> Sequence.create(Sequence.OEIS.Core, :a001157) |> Sequence.take!(10)
[1, 5, 10, 21, 26, 50, 50, 85, 91, 130]
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create_sequence_a005100(opts)

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OEIS Sequence A005100 - Deficient Numbers

From OEIS A005100:

Deficient numbers: numbers n such that sigma(n) < 2n. (Formerly M0514)

Sequence IDs: :a005100

Finite: False

Offset: 1

Example

iex> Sequence.create(Sequence.OEIS.Core, :a005100) |> Sequence.take!(25)
[1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32]
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create_sequence_a005101(opts)

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OEIS Sequence A005101 - Abundant Numbers

From OEIS A005101:

Abundant numbers (sum of divisors of n exceeds 2n). (Formerly M4825)

Sequence IDs: :a005101

Finite: False

Offset: 1

Example

iex> Sequence.create(Sequence.OEIS.Core, :a005101) |> Sequence.take!(10)
[12, 18, 20, 24, 30, 36, 40, 42, 48, 54]