A368978 The number of bi-unitary divisors of n that are squares (A000290).

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, 3, 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, Jan 11 2024

Keywords

• nonn
• easy
• mult

Comments

First differs from A007424, A278908, A307848, A323308, A358260 and A365549 at n = 32.

Links

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

Crossrefs

Cf. A000005, A000290, A005117, A013662, A062503, A222266, A286324, A368977.

Cf. A007424, A278908, A307848, A323308, A365549.

Similar sequences: A046951, A056624, A358260, A368980.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], (e + 1)/2, 2*Floor[(e+2)/4]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = vecprod(apply(x -> if(x%2, (x+1)/2, 2*((x+2)\4)), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = (e + 1)/2 if e is odd, and 2*floor((e+2)/4) if e is even.

a(n) >= 1, with equality if and only if n is squarefree (A005117).

a(n) <= A286324(n), with equality if and only if n is in A062503.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4 + 1/p^5) = 1.58922450321701775833... .