A356241 a(n) is the number of distinct Fermat numbers dividing n.

0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1

Offset: 1

Author

Amiram Eldar, Jul 30 2022

Keywords

• nonn

Comments

A051179(n) is the least number k such that a(k) = n.

The asymptotic density of occurrences of 0 is 1/2.

The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/2^(2^k) = (1/2) * A007404 = 0.4082107545... .

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Fermat Number.
• Wikipedia, Fermat number.

Crossrefs

Cf. A000215, A007404, A051158, A051179, A356242.

Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).

Programs

Mathematica

f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Count[f, _?(Divisible[n, #] &)]; Array[a, 100]

Formula

a(A000215(n)) = 1.

a(A051179(n)) = n.

a(A003593(n)) = A112753(n).

a(n) <= A356242(n).

a(A080307(n)) > 0 and a(A080308(n)) = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(2^(2^k)+1) = 0.5960631721... (A051158).