A368168 The number of unitary divisors of n that are cubefull exponentially odd numbers (A335988).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

1,8

Comments

First differs from A359411 and A367516 at n = 64.

Also, the number of unitary divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

Links

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

Formula

a(n) = A034444(A368167(n)).

Multiplicative with a(p^e) = 2 if e is odd that is larger than 1, and 1 otherwise.

a(n) >= 1, with equality if and only if n is in A335275.

a(n) <= n, with equality if and only if n is in A335988.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 1.12560687309375943599... .

Mathematica

f[p_, e_] := If[e == 1 || EvenQ[e], 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, 2)); }

Crossrefs

Cf. A013661, A034444, A055076, A325837, A335275, A335988, A368167, A368169.

Cf. A359411, A367516.

Sequence in context: A307427 A318672 A359910 * A359411 A367516 A368979

Adjacent sequences: A368165 A368166 A368167 * A368169 A368170 A368171

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 14 2023

Status

approved