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

1,4

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

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... .

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]

Prog

(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])))};

Crossrefs

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

Similar sequences: A046951, A056624, A056626.

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

Sequence in context: A365549 A278908 A307848 * A368978 A255326 A085424

Adjacent sequences: A358257 A358258 A358259 * A358261 A358262 A358263

Keyword

nonn,mult

Author

Amiram Eldar, Nov 06 2022

Status

approved