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A372739 a(n) is the number of possible values of k such that the sum of aliquot coreful divisors of k (A336563) is n.

%I A372739 #14 Apr 26 2025 20:27:56
%S A372739 0,1,1,0,1,3,1,0,0,2,1,1,1,3,2,0,1,1,1,0,2,2,1,1,0,2,0,0,1,6,1,0,2,2,
%T A372739 2,1,1,2,3,0,1,5,1,0,0,2,1,0,0,0,2,0,1,0,2,1,2,2,1,2,1,3,0,0,2,4,1,0,
%U A372739 2,4,1,0,1,2,0,0,2,5,1,1,0,2,1,1,2,2,2
%N A372739 a(n) is the number of possible values of k such that the sum of aliquot coreful divisors of k (A336563) is n.
%C A372739 A coreful divisor d of n is a divisor that is divisible by every prime that divides n (see also A307958).
%H A372739 Amiram Eldar, <a href="/A372739/b372739.txt">Table of n, a(n) for n = 1..10000</a>
%F A372739 a(n) = 0 if and only if n is in A372740.
%F A372739 a(n) = 1 if and only if n is in A372742.
%e A372739 a(2) = 1 since there is 1 possible value of k, k = 4, such that A336563(k) = 2.
%e A372739 a(6) = 3 since there are 3 possible values of k, k = 8, 12 and 18, such that A336563(k) = 6.
%t A372739 f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; seq[max_] := Module[{v = Table[0, {max}], i}, Do[i = s[k]; If[0 < i <= max, v[[i]]++], {k, 1, max^2}]; v]; seq[100]
%o A372739 (PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1) - n;}
%o A372739 lista(nmax) = {my(v = vector(nmax), i); for(k = 1, nmax^2, i = s(k); if(i > 0 && i <= nmax, v[i]++)); v;}
%Y A372739 Cf. A307958, A336563, A372740, A372742.
%Y A372739 Similar sequences: A048138, A324938, A331971, A331973.
%K A372739 nonn
%O A372739 1,6
%A A372739 _Amiram Eldar_, May 12 2024