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 1-4 of 4 results.

A280169 Expansion of Product_{k>=2} 1/(1 - mu(2*k-1)^2*x^(2*k-1)), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 8, 9, 10, 11, 13, 14, 17, 18, 21, 24, 26, 30, 33, 38, 42, 47, 53, 58, 65, 73, 80, 90, 99, 110, 122, 134, 149, 164, 181, 199, 220, 242, 266, 292, 321, 352, 386, 424, 463, 507, 554, 606, 662, 722, 788, 860, 936, 1020, 1111, 1208, 1314, 1428, 1553, 1685, 1829, 1984, 2152
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 27 2016

Keywords

Comments

Number of partitions of n into odd squarefree parts > 1.

Examples

			a(13) = 3 because we have [13], [7, 3, 3] and [5, 5, 3].

Links

Crossrefs

Cf. A005117, A008683, A056911, A073576, A134345, A144338, A280127.

Programs

  • Mathematica
    nmax = 76; CoefficientList[Series[Product[1/(1 - MoebiusMu[2 k - 1]^2 x^(2 k - 1)), {k, 2, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=2} 1/(1 - mu(2*k-1)^2*x^(2*k-1)).

A280198 Expansion of 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)), where mu() is the Moebius function (A008683).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 33, 53, 86, 138, 222, 357, 574, 923, 1484, 2387, 3839, 6173, 9927, 15964, 25672, 41284, 66389, 106762, 171686, 276091, 443989, 713988, 1148179, 1846411, 2969252, 4774918, 7678647, 12348195, 19857396, 31933099, 51352294, 82580715, 132799801, 213558181, 343427445, 552272966, 888121883, 1428207656
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 28 2016

Keywords

Comments

Number of compositions (ordered partitions) into odd squarefree parts (A056911).

Examples

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

Links

Crossrefs

Cf. A005117, A008683, A056911, A134345, A280194.

Programs

  • Mathematica
    nmax = 46; CoefficientList[Series[1/(1 - Sum[MoebiusMu[2 k - 1]^2 x^(2 k - 1), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

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

A331982 Number of compositions (ordered partitions) of n into distinct odd squarefree parts.

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 2, 1, 4, 6, 2, 7, 4, 7, 4, 13, 30, 13, 8, 25, 32, 31, 56, 37, 82, 42, 104, 168, 128, 175, 152, 181, 226, 307, 252, 439, 326, 691, 372, 943, 1190, 1069, 1238, 1435, 2056, 1806, 2102, 2185, 3664, 2550, 4480, 3175, 6090, 3781, 7628, 9691
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 03 2020

Keywords

Examples

			a(8) = 4 because we have [7, 1], [5, 3], [3, 5] and [1, 7].

Links

Crossrefs

Cf. A056911, A087188, A134337, A134345, A280194, A280198, A331846, A331981.

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 1-4 of 4 results.

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