A325801 Number of divisors of n minus the number of distinct positive subset-sums of the prime indices of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, May 23 2019

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with sum A056239(n). A positive subset-sum of an integer partition is any sum of a nonempty submultiset of it.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Positions of 0's are A299702.

Positions of 1's are A325802.

Positions of positive integers are A299729.

Cf. A000005, A002033, A056239, A108917, A112798, A276024, A304793.

Cf. A325694, A325780, A325781, A325792, A325793, A325799.

Programs

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
Table[DivisorSigma[0,n]-Length[Union[hwt/@Divisors[n]]],{n,100}]

PARI

A325801(n) = (numdiv(n) - A299701(n));
A299701(n) = { my(f = factor(n), pids = List([])); for(i=1,#f~, while(f[i,2], f[i,2]--; listput(pids,primepi(f[i,1])))); pids = Vec(pids); my(sv=vector(vecsum(pids))); for(b=1,(2^length(pids))-1,sv[sumbybits(pids,b)] = 1); 1+vecsum(sv); }; \\ Not really an optimal way to count these.
sumbybits(v,b) = { my(s=0,i=1); while(b>0,s += (b%2)*v[i]; i++; b >>= 1); (s); }; \\ Antti Karttunen, May 26 2019

Formula

a(n) = A000005(n) - A299701(n).