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.

A280152 Expansion of Product_{k>=1} (1 + floor(1/omega(2*k+1))*x^(2*k+1)), where omega() is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 3, 3, 3, 3, 4, 4, 5, 4, 6, 6, 6, 7, 7, 9, 8, 9, 10, 11, 12, 11, 14, 14, 16, 15, 18, 19, 19, 21, 22, 25, 25, 27, 28, 32, 32, 34, 36, 40, 41, 42, 47, 49, 52, 53, 57, 62, 63, 67, 71, 76, 79, 82, 88, 93, 98, 100, 108, 114, 118, 124
Offset: 0

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Author

Ilya Gutkovskiy, Dec 27 2016

Keywords

Comments

Number of partitions of n into distinct odd prime powers (1 excluded).

Examples

			a(16) = 3 because we have [13, 3], [11, 5], [9, 7].

Links

Crossrefs

Cf. A001221, A054685, A061345, A246655, A280151.

Programs

  • Mathematica
    nmax = 78; CoefficientList[Series[Product[1 + Floor[1/PrimeNu[2 k + 1]] x^(2 k + 1), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + floor(1/omega(2*k+1))*x^(2*k+1)).

A280200 Expansion of 1/(1 - Sum_{k>=2} floor(1/omega(2*k-1))*x^(2*k-1)), where omega() is the number of distinct prime factors (A001221).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 16, 21, 26, 37, 47, 61, 84, 108, 143, 191, 249, 331, 437, 575, 763, 1004, 1326, 1754, 2311, 3055, 4036, 5323, 7033, 9288, 12257, 16193, 21379, 28223, 37278, 49212, 64984, 85815, 113297, 149614, 197551, 260839, 344439, 454795, 600517, 792958, 1047023, 1382519, 1825533, 2410456, 3182845
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2016

Keywords

Comments

Number of compositions (ordered partitions) into odd prime powers (1 excluded).

Examples

			a(10) = 3 because we have [7, 3], [5, 5] and [3, 7].

Links

Crossrefs

Cf. A001221, A023359, A061345, A246655, A280151, A280195.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[1/(1 - Sum[Floor[1/PrimeNu[2 k - 1]] x^(2 k - 1), {k, 2, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=2} floor(1/omega(2*k-1))*x^(2*k-1)).

A352165 Number of partitions of n into odd prime powers (1 included).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 26, 31, 37, 44, 52, 61, 71, 83, 97, 112, 130, 150, 173, 199, 228, 261, 298, 340, 386, 439, 497, 563, 637, 718, 809, 910, 1023, 1147, 1286, 1439, 1608, 1796, 2003, 2231, 2483, 2761, 3065, 3401, 3770, 4175, 4619
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 06 2022

Keywords

Crossrefs

Cf. A000961, A023893, A023894, A061345, A099773, A134345, A280151, A352166.

Programs

  • Mathematica
    nmax = 55; CoefficientList[Series[Product[1/(1 - Boole[(PrimePowerQ[k] || k == 1) && OddQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=0} 1 / (1 - x^A061345(k)).
Showing 1-3 of 3 results.

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

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