A358261 a(n) is the number of noninfinitary square divisors of n.

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, 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, 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

Offset

1

Comments

First differs from A295884 at n = 64.

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

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... .

Mathematica

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]

Prog

(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])))};

Crossrefs

Cf. A037445, A077609, A295884, A358262.

Similar sequences: A046951, A056624, A056626, A358260.

Sequence in context: A023974 A277164 A011730 * A295884 A101637 A337380

Adjacent sequences: A358258 A358259 A358260 * A358262 A358263 A358264

Keyword

nonn

Author

Amiram Eldar, Nov 06 2022

Status

approved