cp's OEIS Frontend

A368251 The number of nonsquarefree divisors of n that are powers of squarefree numbers (A072777).

%I A368251 #11 Oct 02 2024 10:54:50
%S A368251 0,0,0,1,0,0,0,2,1,0,0,1,0,0,0,3,0,1,0,1,0,0,0,2,1,0,2,1,0,0,0,4,0,0,
%T A368251 0,3,0,0,0,2,0,0,0,1,1,0,0,3,1,1,0,1,0,2,0,2,0,0,0,1,0,0,1,5,0,0,0,1,
%U A368251 0,0,0,4,0,0,1,1,0,0,0,3,3,0,0,1,0,0,0
%N A368251 The number of nonsquarefree divisors of n that are powers of squarefree numbers (A072777).
%C A368251 First differs from A046660 and A066301 at n = 36, and from A183094 at n = 72.
%C A368251 Let b(n, k) be the sequence that counts the divisors of n that are k-th powers of squarefree numbers. Then, b(n, 1) = A034444(n), b(n, 2) = A323308(n), b(n, 3) = A368248(n). b(n, k) is multiplicative with b(p^e, k) = 2 if e >= k, and 1 otherwise. The asymptotic mean of b(n, k) for k >= 2 is lim_{m->oo} (1/m) * Sum_{n=1..m} b(n, k) = zeta(k)/zeta(2*k). Since a(n) = Sum_{k>=2} (b(n, k) - 1), the formula for the asymptotic mean of this sequence follows (see the Formula section).
%H A368251 Amiram Eldar, <a href="/A368251/b368251.txt">Table of n, a(n) for n = 1..10000</a>
%F A368251 a(n) = A327527(n) - A034444(n).
%F A368251 a(n) = 0 if and only if n is squarefree (A005117).
%F A368251 Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) = 0.848633... (A368250).
%t A368251 a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1] - 2^Length[e]]; a[1] = 0; Array[a, 100]
%o A368251 (PARI) a(n) = {my(f = factor(n), e, m, h, c); if(n == 1, 0, e = f[,2]; m = vecmax(e); h = vector(m); for(i = 1,m, c = 0; for(j = 1, #e, if(e[j] == (m+1-i), c++)); h[i] = c); for(i = 2, m, h[i] += h[i-1]); for(i = 1, m, h[i] = 2^h[i]-1); 1 + vecsum(h) - 1<<#e);}
%Y A368251 Cf. A005117, A048105, A072777, A327527, A368250.
%Y A368251 Cf. A034444, A323308, A368248.
%Y A368251 Cf. A046660, A066301, A183094.
%K A368251 nonn
%O A368251 1,8
%A A368251 _Amiram Eldar_, Dec 19 2023