A358261 - OEIS

%I #9 Nov 07 2022 02:10:38

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0

%N a(n) is the number of noninfinitary square divisors of n.

%C First differs from A295884 at n = 64.

%C The first occurrence of 1 is at n = 16, of 2 is at n = 144, of 3 is at n = 256, ... (see A358262).

%H Amiram Eldar, <a href="/A358261/b358261.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A046951(n) - A358260(n).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/6 - Product_{p prime} ((1-1/p) * Sum_{k>=1} a(p^k)/p^k) = 0.09038522017381649992... .

%t f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n));  prod(i=1, #f~, 1+f[i,2]\2) - prod(i=1, #f~, 2^hammingweight(if(f[i,2]%2, f[i,2]-1, f[i,2])))};

%Y Cf. A037445, A077609, A295884, A358262.

%Y Similar sequences: A046951, A056624, A056626, A358260.

%K nonn

%O 1

%A _Amiram Eldar_, Nov 06 2022