A014652 - OEIS

%I #30 Sep 30 2018 02:36:02

%S 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,

%T 226,1,16,23,26,23,120,1,29,27,92,1,422,1,56,78,35,1,208,7,140,35,66,

%U 1,261,35,128,39,44,1,1487,1,47,108,32,41,996,1,86,47,1062,1,456,1,56

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

%H Antti Karttunen, <a href="/A014652/b014652.txt">Table of n, a(n) for n = 1..10000</a>

%H David A. Corneth, <a href="/A014652/a014652.txt">PARI program</a>

%F 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

%o (PARI)

%o \\ This is for computing just a moderate number of terms:

%o prime_factors_with1_reversed(n) = vecsort(setunion([1],factor(n)[,1]~), , 4);

%o 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)));

%o A014652(n) = partitions_into_with_trailing_ones(n-1,prime_factors_with1_reversed(n)); \\ _Antti Karttunen_, Sep 10 2018

%o (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

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

%K nonn

%O 1,4

%A _Olivier Gérard_