A358260 a(n) is the number of infinitary square divisors of n.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Nov 06 2022

Keywords

• nonn
• mult

Comments

First differs from A007424 at n = 36, from A323308 at n = 64, and from A278908 and A307848 at n = 128.

Links

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

Crossrefs

Cf. A000120, A048881, A037445, A077609, A358261.

Similar sequences: A046951, A056624, A056626.

Sequences with the same initial terms: A007424, A278908, A307848, A323308.

Programs

Mathematica

f[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

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

Formula

Multiplicative with a(p^e) = 2^A000120(e) if e is even, and 2^A000120(e-1) if e is odd.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((1-1/p) * Sum_{k>=1} a(p^k)/p^k) = 1.55454884667440993654... .