A014649 -id:A014649 - cp's OEIS frontend

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

Olivier Gérard

nonn

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

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

\\ 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

\\ 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

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

A014650 Number of partitions of n into its divisors that are powers of primes (A000961) with at least one part of size 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 1, 6, 3, 8, 1, 27, 1, 11, 11, 26, 1, 43, 1, 63, 15, 17, 1, 215, 5, 20, 18, 114, 1, 226, 1, 166, 23, 26, 23, 734, 1, 29, 27, 728, 1, 422, 1, 261, 181, 35, 1, 2697, 7, 179, 35, 357, 1, 791, 35, 1729, 39, 44, 1, 6747, 1, 47, 325, 1626, 41, 996, 1, 594, 47, 1062, 1, 20345, 1, 56, 327, 735, 47, 1374, 1, 13485, 216, 62, 1

Offset: 1

Olivier Gérard

nonn

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

Cf. A000961, A014648, A014649, A014651, A014652, A066874.

\\ This is for computing a small number of terms:
primepower_divisors_with1_reversed(n) = vecsort(select(d -> ((1==d) || isprimepower(d)), divisors(n)), , 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)));
A014650(n) = partitions_into_with_trailing_ones(n-1,primepower_divisors_with1_reversed(n)); \\ Antti Karttunen, Sep 10 2018

\\ 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

More terms from and the name clarified by Antti Karttunen, Sep 10 2018

cp's OEIS Frontend

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

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A014650 Number of partitions of n into its divisors that are powers of primes (A000961) with at least one part of size 1.

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PARI

PARI

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