A286852 Number of partitions of n into unitary prime divisors of n.

1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 21, 1, 0, 2, 2, 2, 0, 1, 2, 2, 1, 1, 28, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 5, 1, 2, 1, 0, 2, 42, 1, 1, 2, 43, 1, 0, 1, 2, 1, 1, 2, 49, 1, 1, 0, 2, 1, 5, 2, 2, 2, 1, 1, 10, 2, 1, 2, 2, 2

Offset

0,7

Links

Antti Karttunen, Table of n, a(n) for n = 0..2309

Eric Weisstein's World of Mathematics, Unitary Divisor

Index entries for related partition-counting sequences

Formula

a(n) = [x^n] Product_{p|n, p prime, gcd(p, n/p) = 1} 1/(1 - x^p).

a(n) = 0 if n is a powerful number (A001694).

Example

a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are unitary prime divisors {2, 3} therefore we have [3, 3] and [2, 2, 2].

Mathematica

Join[{1}, Table[d = Divisors[n]; Coefficient[Series[Product[1/(1 - Boole[GCD[n/d[[k]], d[[k]]] == 1 && PrimeQ[d[[k]]]] x^d[[k]]), {k, Length[d]}], {x, 0, n}], x, n], {n, 1, 95}]]

Prog

(PARI)
A055231(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k, 2] > 1, f[k, 2] = 0); ); factorback(f); } \\ From A055231
unitary_prime_factors(n) = { my(ufs = factor(A055231(n))); ufs[, 1]~; };
partitions_into(n, parts, from=1) = if(!n, 1, my(k = #parts, s=0); for(i=from, k, if(parts[i]<=n, s += partitions_into(n-parts[i], parts, i))); (s));
A286852(n) = if(n<2, 1-n, partitions_into(n, vecsort(unitary_prime_factors(n), , 4))); \\ Antti Karttunen, Jul 02 2018

Crossrefs

Cf. A018818, A055231, A056169, A066874, A066882, A225244, A284289.

Sequence in context: A335452 A355817 A056169 * A341595 A369428 A125070

Adjacent sequences: A286849 A286850 A286851 * A286853 A286854 A286855

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 01 2017

Status

approved