A380089 The number of unitary divisors of n that are terms in A207481.

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 2, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 4, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4

Offset: 1

Author

Amiram Eldar, Jan 12 2025

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A005117, A054743, A077610, A185359, A207481, A377519, A380086, A380088, A380090.

Programs

Mathematica

f[p_, e_] := If[e <= p, 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~,if(f[i,2] <= f[i,1], 2, 1));}

Formula

a(n) = A034444(A380088(n)).

Multiplicative with a(p^e) = 2 if e <= p, and 1 otherwise.

a(n) = 1 if and only if n is in A054743.

a(n) < A034444(n) if and only if n is in A185359.

a(n) = A034444(n) if and only if n is in A207481.

a(n) = A377519(n) if and only if n is squarefree (A005117).

Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^((p+1)*s)).

Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A034444(k) = Product_{p prime} (1 - 1/(2*p^(p+1))) = 0.93168306734008028353...