A366309 The number of infinitary divisors of n that are terms of A366243.

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

Offset: 1

Author

Amiram Eldar, Oct 06 2023

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A037445, A139352, A366242, A366243, A366245, A366247, A366246, A366308.

Programs

Mathematica

s[0] = 0; s[n_] := s[n] = s[Floor[n/4]] + If[Mod[n, 4] > 1, 1, 0]; f[p_, e_] := 2^s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

s(e) = if(e > 3, s(e\4)) + e%4\2 \\ after Charles R Greathouse IV at A139352
a(n) = vecprod(apply(x -> 2^s(x), factor(n)[, 2]));

Formula

Multiplicative with a(p^e) = 2^A139352(e).

a(n) = 2^A366247(n).

a(n) = A037445(n)/A366308(n).

a(n) = A037445(A366245(n)).

a(n) >= 1, with equality if and only if n is in A366242.

a(n) <= A037445(n), with equality if and only if n is in A366243.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 - 1/p)*(1 + Sum_{k>=1} 2^A139352(k)/p^k) = 1.44736831993091923328... .