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%I #7 May 28 2019 19:39:23
%S 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,
%T 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,
%U 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
%N Number of divisors of n minus the number of distinct positive subset-sums of the prime indices of n.
%C 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.
%H Antti Karttunen, <a href="/A325801/b325801.txt">Table of n, a(n) for n = 1..20000</a>
%F a(n) = A000005(n) - A299701(n).
%t hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p] k]];
%t Table[DivisorSigma[0,n]-Length[Union[hwt/@Divisors[n]]],{n,100}]
%o (PARI)
%o A325801(n) = (numdiv(n) - A299701(n));
%o 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.
%o 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
%Y Positions of 0's are A299702.
%Y Positions of 1's are A325802.
%Y Positions of positive integers are A299729.
%Y Cf. A000005, A002033, A056239, A108917, A112798, A276024, A304793.
%Y Cf. A325694, A325780, A325781, A325792, A325793, A325799.
%K nonn
%O 1,24
%A _Gus Wiseman_, May 23 2019
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