A014652 Number of partitions of n in its prime divisors with at least one part of size 1.

1, 1, 1, 2, 1, 5, 1, 4, 3, 8, 1, 16, 1, 11, 11, 8, 1, 33, 1, 26, 15, 17, 1, 56, 5, 20, 9, 36, 1, 226, 1, 16, 23, 26, 23, 120, 1, 29, 27, 92, 1, 422, 1, 56, 78, 35, 1, 208, 7, 140, 35, 66, 1, 261, 35, 128, 39, 44, 1, 1487, 1, 47, 108, 32, 41, 996, 1, 86, 47, 1062, 1, 456, 1, 56

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• David A. Corneth, PARI program

Crossrefs

Cf. A014648, A014649, A014650, A014651, A066874, A066882, A286852.

Programs

PARI

\\ This is for computing just a moderate number of terms:
prime_factors_with1_reversed(n) = vecsort(setunion([1],factor(n)[,1]~), , 4);
partitions_into_with_trailing_ones(n,parts,from=1) = if(!n,1, if(#parts<=(from+1), if(#parts == from,1,(1+(n\parts[from]))), my(s=0); for(i=from,#parts,if(parts[i]<=n, s += partitions_into_with_trailing_ones(n-parts[i],parts,i))); (s)));
A014652(n) = partitions_into_with_trailing_ones(n-1,prime_factors_with1_reversed(n)); \\ Antti Karttunen, Sep 10 2018

PARI

\\ For an efficient program to compute large numbers of terms, see David A. Corneth's PARI program included in the Links section. - Antti Karttunen, Sep 12 2018

Formula

Coefficient of x^(n-1) in expansion of (1/(1-x))*1/Product_{d is prime divisor of n} (1-x^d). - Vladeta Jovovic, Apr 11 2004