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

1,4

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

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... .

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]

Prog

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

Crossrefs

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

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

Similar sequences: A046951, A056624, A358260, A368980.

Sequence in context: A278908 A307848 A358260 * A255326 A085424 A088737

Adjacent sequences: A368975 A368976 A368977 * A368979 A368980 A368981

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jan 11 2024

Status

approved