A321648 Number of permutations of the conjugate of the integer partition with Heinz number n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 3, 1, 1, 2, 1, 3, 6, 5, 1, 2, 1, 6, 1, 4, 1, 6, 1, 1, 10, 7, 4, 2, 1, 8, 15, 3, 1, 12, 1, 5, 3, 9, 1, 2, 1, 3, 21, 6, 1, 2, 10, 4, 28, 10, 1, 6, 1, 11, 6, 1, 20, 20, 1, 7, 36, 12, 1, 2, 1, 12, 3, 8, 5, 30, 1, 3, 1, 13

Offset

1,6

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = A008480(A122111(n)).

Example

The a(42) = 12 permutations: (3211), (3121), (3112), (2311), (2131), (2113), (1321), (1312), (1231), (1213), (1132), (1123).

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Permutations[conj[primeMS[n]]]], {n, 50}]

Prog

(PARI)
A008480(n) = {my(sig=factor(n)[, 2]); vecsum(sig)!/factorback(apply(k->k!, sig))}; \\ From A008480
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)};
A122111(n) = if(1==n, n, prime(bigomega(n))*A122111(A064989(n)));
A321648(n) = A008480(A122111(n)); \\ Antti Karttunen, Dec 23 2018

Crossrefs

Cf. A008480, A056239, A112798, A122111, A296150, A321645, A321646, A321647, A321649, A321650.

Sequence in context: A073408 A372572 A120454 * A330755 A316431 A319567

Adjacent sequences: A321645 A321646 A321647 * A321649 A321650 A321651

Keyword

nonn

Author

Gus Wiseman, Nov 15 2018

Status

approved