A325801 - OEIS

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

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OFFSET

1,24

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

FORMULA

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

MATHEMATICA

hwt[n_]:=Total[Cases[FactorInteger[n], {p_, k_}:>PrimePi[p] k]];

Table[DivisorSigma[0, n]-Length[Union[hwt/@Divisors[n]]], {n, 100}]

PROG

(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

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.

Sequence in context: A122841 A326075 A060862 * A325194 A066087 A294927

Adjacent sequences: A325798 A325799 A325800 * A325802 A325803 A325804

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 23 2019

STATUS

approved