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.

Previous Showing 61-64 of 64 results.

A321936 Number of integer partitions of n containing no 1's, prime powers, or squarefree numbers.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 0, 3, 0, 2, 0, 3, 1, 1, 0, 7, 0, 2, 0, 5, 0, 5, 0, 7, 1, 3, 0, 12, 0, 4, 2, 10, 1, 8, 0, 14, 2, 6, 0, 22, 1, 10, 3, 20, 1, 15, 0, 26, 5, 12, 2
Offset: 0

Views

Author

Gus Wiseman, Dec 11 2018

Keywords

Comments

Number of integer partitions of n using elements of A126706.

Examples

			The a(56) = 7 partitions:
  (56)
  (28,28)
  (36,20)
  (44,12)
  (20,18,18)
  (24,20,12)
  (20,12,12,12)

Crossrefs

Cf. A000607, A000961, A002095, A005117, A023893, A023894, A078135, A114374, A126706, A246655, A321346, A321347, A321378, A321665.

Programs

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

A328970 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j) / (1 - x^prime(j)) is zero.

Original entry on oeis.org

2, 3, 9, 11, 12, 14, 17, 18, 19, 20, 28, 44, 47, 51, 52, 55, 56, 58, 59, 62, 64, 65, 69, 80, 81, 82, 83, 87, 91, 92, 94, 96, 99, 105, 106, 107, 113, 118, 119, 126, 127, 131, 147, 155, 157, 160, 161, 162, 164, 178, 179, 180, 215, 218, 224, 227, 257, 259, 269, 295
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 01 2019

Keywords

Comments

Numbers k such that number of partitions of k into an even number of distinct nonprime parts equals number of partitions of k into an odd number of distinct nonprime parts.
Positions of 0's in A302234.

Crossrefs

Cf. A002095, A018252, A090864, A096258, A302234.

Programs

  • Mathematica
    a[j_] := a[j] = If[j == 0, 1, -Sum[Sum[Boole[!PrimeQ[d]] d, {d, Divisors[k]}] a[j - k], {k, 1, j}]/j]; Select[Range[300], a[#] == 0 &]
Flatten[Position[nmax = 300; Rest[CoefficientList[Series[Product[(1 - x^j)/(1 - x^Prime[j]), {j, 1, nmax}], {x, 0, nmax}], x]], 0]]

A339219 Number of partitions of n into nonprime parts where every part appears at least 2 times.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 4, 2, 4, 4, 6, 4, 8, 6, 11, 8, 11, 11, 17, 11, 19, 18, 25, 20, 32, 26, 42, 32, 46, 43, 63, 47, 72, 66, 90, 74, 110, 94, 137, 115, 155, 145, 203, 161, 235, 212, 283, 244, 339, 298, 413, 356, 472, 437, 589, 496, 681, 625, 810, 718, 962
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 27 2020

Keywords

Examples

			a(12) = 4 because we have [6, 6], [4, 4, 4], [4, 4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A002095, A007690, A018252, A161077, A339218.

Programs

Formula

G.f.: Product_{k>=1} (1 + x^(2*A018252(k)) / (1 - x^A018252(k))).

A368870 Number of partitions of n into nonprime parts not greater than sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 37, 37, 40, 40, 44, 44, 48, 48, 52, 52, 56, 56, 61, 61, 65, 65, 70, 70, 75, 75, 80, 80, 85, 85, 91, 91, 96, 96, 351, 351, 378, 378, 411, 411, 441
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 08 2024

Keywords

Crossrefs

Cf. A002095, A097356.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(isprime(i), 0, b(n-i, min(n-i, i)))))
    end:
a:= n-> b(n, floor(sqrt(n))):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 13 2024
  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - Boole[!PrimeQ[k]] x^k), {k, 1, Floor[Sqrt[n]]}], {x, 0, n}], {n, 0, 70}]
Previous Showing 61-64 of 64 results.

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

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