A304795 - OEIS

%I #13 Jul 03 2018 03:24:31

%S 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,

%T 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,

%U 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

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

%C 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).

%H Antti Karttunen, <a href="/A304795/b304795.txt">Table of n, a(n) for n = 1..65537</a>

%e 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.

%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]],Total],Length[#]===1&];

%t Table[Length[uqsubs[primeMS[n]]],{n,100}]

%o (PARI)

%o up_to = 65537;

%o A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }

%o v056239 = vector(up_to,n,A056239(n));

%o 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

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

%K nonn

%O 1,4

%A _Gus Wiseman_, May 18 2018

%E More terms from _Antti Karttunen_, Jul 02 2018