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A327166 Number of divisors d of n for which A000005(d)*d is equal to n, where A000005(x) gives the number of divisors of x.

%I A327166 #10 Jul 23 2022 04:15:03
%S A327166 1,0,0,1,0,1,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,1,0,1,0,1,1,0,0,0,0,1,0,1,
%T A327166 0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,0,0,
%U A327166 0,0,0,1,0,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,2,0,0,0,0,0,0,0,0,0,1,0,1
%N A327166 Number of divisors d of n for which A000005(d)*d is equal to n, where A000005(x) gives the number of divisors of x.
%C A327166 a(n) tells how many times in total n occurs in A038040.
%H A327166 Antti Karttunen, <a href="/A327166/b327166.txt">Table of n, a(n) for n = 1..65537</a>
%F A327166 a(n) = Sum_{d|n} [A000005(d)*d == n], where [ ] is the Iverson bracket.
%e A327166 108 has the following twelve divisors: [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108]. Of these, only d=18 and d=27 are such that d*A000005(d) = 108, as 18*6 = 27*4 = 108. Thus a(108) = 2.
%t A327166 Table[Sum[If[d*DivisorSigma[0, d] == n, 1, 0], {d, Divisors[n]}], {n, 1, 120}] (* _Vaclav Kotesovec_, Jul 23 2022 *)
%o A327166 (PARI) A327166(n) = sumdiv(n,d,(d*numdiv(d))==n);
%Y A327166 Cf. A000005, A038040.
%Y A327166 Cf. also A327153, A327169.
%K A327166 nonn
%O A327166 1,108
%A A327166 _Antti Karttunen_, Sep 19 2019