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.

A321347 Number of strict integer partitions of n containing no prime powers (including 1).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 4, 4, 2, 3, 4, 4, 5, 6, 5, 6, 7, 7, 9, 10, 10, 13, 12, 11, 15, 17, 16, 19, 20, 20, 25, 28, 26, 30, 33, 35, 41, 43, 42, 50, 55, 57, 64, 67, 67, 79, 86, 87, 97, 105, 109, 124, 131, 135, 151, 163, 169
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Comments

First differs from A286221 at a(30) = 6, A286221(30) = 5.

Examples

			The a(36) = 13 strict integer partitions:
  (36),
  (21,15), (22,14), (24,12), (26,10), (30,6), (35,1),
  (14,12,10), (18,12,6), (20,10,6), (20,15,1), (21,14,1),
  (15,14,6,1).

Links

Crossrefs

Cf. A000607, A000961, A001597, A002095, A023893, A023894, A096258, A246655, A320322, A321346, A321378, A321665, A322452, A322454.

Programs

  • Mathematica
    nn=100;
ser=Product[If[PrimePowerQ[n],1,1+x^n],{n,nn}];
CoefficientList[Series[ser,{x,0,nn}],x]

A286220 Number of partitions of n into distinct parts with an odd number of distinct prime divisors.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 3, 5, 5, 6, 7, 7, 10, 9, 12, 12, 15, 15, 18, 19, 22, 24, 26, 30, 32, 36, 40, 43, 49, 52, 58, 63, 69, 76, 81, 91, 96, 108, 114, 127, 135, 148, 159, 173, 186, 202, 217, 234, 253, 271, 293, 313, 339, 361, 390, 416, 449, 478, 514, 547, 588, 625, 671, 714, 763, 815, 867
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2017

Keywords

Examples

			a(9) = 4 because we have [9], [7, 2], [5, 4] and [4, 3, 2].

Links

Crossrefs

Cf. A030230, A285799, A286221.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[1 + Boole[OddQ[PrimeNu[k]]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A030230(k)).

A286223 Number of partitions of n into distinct parts with an even number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 3, 2, 0, 1, 3, 4, 4, 2, 1, 4, 6, 5, 4, 4, 6, 10, 10, 6, 6, 10, 13, 14, 11, 9, 14, 21, 21, 17, 17, 23, 31, 31, 25, 25, 33, 41, 43, 39, 38, 50, 61, 60, 56, 58, 68, 83, 87, 79, 82, 99, 115, 121, 118, 118, 139, 163, 164, 157, 165, 189, 216, 228, 221, 229, 265, 296
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2017

Keywords

Examples

			a(10) = 3 because we have [10], [9, 1] and [6, 4].

Links

Crossrefs

Cf. A028260, A286219, A286221, A286222.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1 + Boole[EvenQ[PrimeOmega[k]]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A028260(k)).

A286225 Number of compositions (ordered partitions) of n into parts with an even number of distinct prime divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 18, 25, 34, 46, 61, 83, 112, 153, 209, 286, 387, 526, 713, 969, 1317, 1794, 2437, 3312, 4497, 6110, 8302, 11290, 15347, 20865, 28354, 38533, 52361, 71167, 96721, 131464, 178672, 242834, 330020, 448532, 609590, 828511, 1126037, 1530418, 2079977, 2826896, 3841998
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2017

Keywords

Examples

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

Links

Crossrefs

Cf. A030231, A285798, A286221, A286224.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[1/(1 - Sum[Boole[EvenQ[PrimeNu[k]]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - Sum_{k>=1} x^A030231(k)).
Showing 1-4 of 4 results.

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