A304793 Number of distinct positive subset-sums of the integer partition with Heinz number n.

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

Offset

1,4

Comments

A positive integer n is a positive subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

a(n) <= A000005(n).

One less than the number of distinct values obtained when A056239 is applied to all divisors of n. - Antti Karttunen, Jul 01 2018

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Example

The positive subset-sums of (4,3,1) are {1, 3, 4, 5, 7, 8} so a(70) = 6.
The positive subset-sums of (5,1,1,1) are {1, 2, 3, 5, 6, 7, 8} so a(88) = 7.

Mathematica

Table[Length[Union[Total/@Rest[Subsets[Join@@Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]], {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));
A304793(n) = { my(m=Map(), s, k=0); fordiv(n, d, if(!mapisdefined(m, s = v056239[d]), mapput(m, s, s); k++)); (k-1); }; \\ Antti Karttunen, Jul 01 2018

Crossrefs

Cf. A056239, A122768, A276024, A284640, A296150, A299701, A299702, A301855, A301935, A301957, A304792, A304795, A305611.

Sequence in context: A356159 A294926 A079167 * A199570 A350338 A239707

Adjacent sequences: A304790 A304791 A304792 * A304794 A304795 A304796

Keyword

nonn

Author

Gus Wiseman, May 18 2018

Extensions

More terms from Antti Karttunen, Jul 01 2018

Status

approved