cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A302234 Expansion of Product_{k>=1} (1 - x^k)/(1 - x^prime(k)).

Original entry on oeis.org

1, -1, 0, 0, -1, 1, -1, 1, -1, 0, 1, 0, 0, 1, 0, -1, 1, 0, 0, 0, 0, -1, 1, -1, 1, -2, 1, -1, 0, 1, -1, 1, -2, 2, -1, -1, 2, -1, -1, 2, -2, 2, -1, 1, 0, -1, 1, 0, 1, -2, 2, 0, 0, 2, -1, 0, 0, 1, 0, 0, 1, -2, 0, -1, 0, 0, -2, 2, -3, 0, 2, -2, 1, -1, 1, -2, 1, -1, -1, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 03 2018

Keywords

Comments

The difference between the number of partitions of n into an even number of distinct nonprime parts and the number of partitions of n into an odd number of distinct nonprime parts.
Convolution of the sequences A000607 and A010815.

Links

Crossrefs

Cf. A000607, A002095, A010815, A018252, A046675, A096258, A302236.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Product[(1 - x^k)/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 80; CoefficientList[Series[Product[(1 - Boole[!PrimeQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 - x^A018252(k)).

A339395 Number of partitions of n into an even number of nonprime parts.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 4, 6, 7, 8, 10, 13, 14, 19, 20, 26, 29, 36, 40, 51, 56, 70, 76, 96, 105, 129, 143, 172, 192, 231, 254, 308, 339, 402, 447, 529, 586, 691, 764, 896, 993, 1159, 1281, 1493, 1652, 1912, 2114, 2445, 2699, 3110, 3436, 3939, 4356, 4982, 5497, 6280
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(9) = 3 because we have [8, 1], [6, 1, 1, 1] and [4, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A002095, A018252, A027187, A184198, A302236, A339380, A339396.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[(1/2) (Product[(1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}] + Product[(1 + x^Prime[k])/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} (1 - x^prime(k)) / (1 - x^k) + Product_{k>=1} (1 + x^prime(k)) / (1 + x^k)).
a(n) = (A002095(n) + A302236(n)) / 2.

A339396 Number of partitions of n into an odd number of nonprime parts.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 4, 6, 6, 9, 9, 13, 14, 18, 20, 26, 29, 37, 39, 51, 57, 69, 78, 95, 105, 129, 141, 173, 192, 231, 255, 306, 340, 403, 446, 531, 585, 691, 764, 896, 995, 1160, 1279, 1493, 1652, 1911, 2117, 2443, 2700, 3109, 3434, 3941, 4357, 4983, 5496, 6277
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

Examples

			a(9) = 3 because we have [9], [4, 4, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A002095, A018252, A027193, A184199, A302236, A339381, A339395.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[(1/2) (Product[(1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}] - Product[(1 + x^Prime[k])/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} (1 - x^prime(k)) / (1 - x^k) - Product_{k>=1} (1 + x^prime(k)) / (1 + x^k)).
a(n) = (A002095(n) - A302236(n)) / 2.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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