A301855 -id:A301855 - cp's OEIS frontend

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

Gus Wiseman, May 18 2018

nonn

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).

			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.

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

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

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}]

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

More terms from Antti Karttunen, Jul 02 2018

A305613 Numbers whose multiset of prime factors is not knapsack.

Original entry on oeis.org

30, 60, 70, 72, 84, 90, 120, 140, 144, 150, 168, 180, 210, 216, 240, 252, 270, 280, 286, 288, 300, 308, 330, 336, 350, 360, 378, 390, 420, 432, 440, 450, 480, 490, 495, 504, 510, 525, 528, 540, 560, 570, 572, 576, 588, 594, 600, 616, 630, 646, 648, 660, 672

Offset: 1

Gus Wiseman, Jun 06 2018

nonn

A multiset of positive integers is knapsack if every distinct submultiset has a different sum.

			30 = 2 * 3 * 5 is not knapsack because 2 + 3 = 5.

Cf. A027746, A108917, A122768, A275972, A276024, A299701, A299729, A301855, A301900, A304793, A305611.

Select[Range[1000],DivisorSigma[0,#]=!=Length[Union[Total/@Subsets[Join@@Cases[FactorInteger[#],{p_,k_}:>Table[p,{k}]]]]]&]

cp's OEIS Frontend

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

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