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 1-6 of 6 results.

A321346 Number of integer partitions of n containing no prime powers > 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 11, 11, 14, 16, 19, 19, 25, 26, 31, 34, 40, 41, 52, 54, 63, 69, 81, 86, 105, 109, 126, 137, 160, 169, 201, 211, 242, 264, 303, 320, 375, 396, 453, 490, 557, 590, 682, 726, 823, 888, 1002, 1065, 1219
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Comments

First differs from A285798 at a(30) = 52, A285798(30) = 51.

Examples

			The a(20) = 14 integer partitions:
  (20)
  (10,10)
  (14,6)
  (18,1,1)
  (12,6,1,1)
  (6,6,6,1,1)
  (10,6,1,1,1,1)
  (15,1,1,1,1,1)
  (14,1,1,1,1,1,1)
  (12,1,1,1,1,1,1,1,1)
  (6,6,1,1,1,1,1,1,1,1)
  (10,1,1,1,1,1,1,1,1,1,1)
  (6,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

Links

Crossrefs

Cf. A000607, A000961, A001597, A002095, A023893, A023894, A096258, A246655, A320322, A321347, A321378, A321665, A322452, A322454.

Programs

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

A285799 Number of partitions of n into parts with an odd number of distinct prime divisors.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 23, 29, 37, 44, 54, 66, 80, 96, 115, 138, 165, 196, 231, 275, 322, 380, 444, 520, 608, 706, 821, 952, 1102, 1272, 1467, 1688, 1941, 2226, 2549, 2917, 3329, 3798, 4324, 4918, 5587, 6337, 7180, 8125, 9184, 10369, 11695, 13174, 14828, 16671, 18723, 21011, 23551
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2017

Keywords

Examples

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

Links

Crossrefs

Cf. A001156 (number of partitions into parts with an odd number of divisors), A030230, A285798.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[1/(1 - Boole[OddQ[PrimeNu[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

A286219 Number of partitions of n into parts with an even number of prime divisors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 7, 9, 10, 13, 15, 19, 20, 25, 28, 34, 38, 46, 50, 61, 69, 81, 89, 105, 116, 137, 152, 175, 194, 226, 250, 288, 318, 363, 403, 462, 508, 577, 637, 721, 796, 900, 988, 1113, 1228, 1378, 1515, 1696, 1860, 2080, 2287, 2546, 2791, 3106, 3402, 3779
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2017

Keywords

Examples

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

Links

Crossrefs

Cf. A028260, A087153, A285798, A286218.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(bigomega(d)::odd, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, May 04 2017
  • Mathematica
    nmax = 60; CoefficientList[Series[Product[1/(1 - Boole[EvenQ[PrimeOmega[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

A286221 Number of partitions of n into distinct parts with an even number of distinct prime divisors.

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, 5, 6, 7, 9, 10, 10, 12, 11, 11, 15, 16, 15, 17, 18, 19, 23, 26, 25, 27, 30, 33, 37, 38, 39, 46, 50, 52, 57, 59, 61, 71, 77, 78, 84, 91, 97, 107, 114, 120, 131, 139, 147, 163, 172, 180, 197
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2017

Keywords

Examples

			a(21) = 4 because we have [21], [20, 1], [15, 6] and [14, 6, 1].

Links

Crossrefs

Cf. A030231, A285798, A286220.

Programs

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

Formula

G.f.: Product_{k>=1} (1 + x^A030231(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)).

A327670 Sum of divisors of n that have an even number of distinct prime factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 19, 1, 15, 16, 1, 1, 25, 1, 31, 22, 23, 1, 43, 1, 27, 1, 43, 1, 32, 1, 1, 34, 35, 36, 73, 1, 39, 40, 71, 1, 42, 1, 67, 61, 47, 1, 91, 1, 61, 52, 79, 1, 79, 56, 99, 58, 59, 1, 64, 1, 63, 85, 1, 66, 62, 1, 103, 70, 60, 1, 169, 1, 75, 91
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 21 2019

Keywords

Links

Crossrefs

Cf. A000961 (positions of 1's), A000203, A030231, A049060, A285798, A318676, A327669.

Programs

  • Maple
    with(numtheory):
a:= n-> add(`if`(nops(factorset(d))::even, d, 0), d=divisors(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 21 2019
  • Mathematica
    a[n_] := DivisorSum[n, # &, EvenQ[PrimeNu[#]] &]; Table[a[n], {n, 1, 75}]

Formula

G.f.: Sum_{k>=1} A030231(k) * x^A030231(k) / (1 - x^A030231(k)).
L.g.f.: log(B(x)) = Sum_{n>=1} a(n) * x^n / n, where B(x) = g.f. of A285798.
a(n) = A000203(n) - A327669(n).
Showing 1-6 of 6 results.

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