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A304795 Number of positive special sums of the integer partition with Heinz number n. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A294926 A079167

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

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Last modified July 24 13:42 EDT 2021. Contains 346273 sequences. (Running on oeis4.)