A304795 Number of positive special sums of the integer partition with Heinz number n.

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 3, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 3, 2, 3, 3, 5, 1, 5, 1, 5, 3, 3, 3, 4, 1, 3, 3, 5, 1, 7, 1, 5, 5, 3, 1, 3, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 3, 1, 3, 3, 6, 3, 7, 1, 5, 3, 5, 1, 3, 1, 3, 5, 5, 3, 7, 1, 5, 4, 3, 1, 5, 3, 3, 3, 7, 1, 5, 3, 5, 3, 3, 3, 3, 1, 5, 5, 8, 1, 7, 1, 7, 7

Offset

1,4

Comments

A positive special sum of y is a number n > 0 such that exactly one submultiset of y sums to n. 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..65537

Example

The a(36) = 4 special sums are 1, 3, 5, 6, corresponding to the submultisets (1), (21), (221), (2211), with Heinz numbers 2, 6, 18, 36.

Mathematica

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]], Total], Length[#]===1&];
Table[Length[uqsubs[primeMS[n]]], {n, 100}]

Prog

(PARI)
up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
v056239 = vector(up_to, n, A056239(n));
A304795(n) = { my(m=Map(), s, k=0, c); fordiv(n, d, if(!mapisdefined(m, s = v056239[d], &c), mapput(m, s, 1), mapput(m, s, c+1))); sumdiv(n, d, (1==mapget(m, v056239[d])))-1; }; \\ Antti Karttunen, Jul 02 2018

Crossrefs

Cf. A000712, A056239, A108917, A122768, A276024, A284640, A296150, A299701, A299702, A301854, A301855, A301957, A304793, A304796.

Sequence in context: A307720 A191350 A329616 * A036459 A356159 A294926

Adjacent sequences: A304792 A304793 A304794 * A304796 A304797 A304798

Keyword

nonn

Author

Gus Wiseman, May 18 2018

Extensions

More terms from Antti Karttunen, Jul 02 2018

Status

approved