A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 0, 2, 0, 1, 2, 0, 1, 1, 0, 0, 1, 2, 0, 3, 0, 0, 2, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 3, 0, 0, 1, 0, 2, 2, 0, 0, 3, 1, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 2, 0, 0, 3, 0, 0, 1, 0, 1, 4, 0, 0, 1, 2, 0, 1

Offset

1,9

Comments

The multiplicity of a divisor d (not necessarily a prime) of n is defined in A169594 (see also the first formula).

A000244(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) = (1/2) * A051158 = 0.2980315860... .

Links

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

Eric Weisstein's World of Mathematics, Fermat Number.

Wikipedia, Fermat number.

Formula

a(n) = Sum_{k>=1} v(A000215(k), n), where v(m, n) is the exponent of the largest power of m that divides n.

a(A000215(n)) = 1.

a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = n.

a(A003593(n)) = A112754(n).

a(n) >= A356241(n).

a(A051179(n)) = 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)) = 0.8164215090... (A007404).

Mathematica

f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Total[IntegerExponent[n, f]]; Array[a, 100]

Crossrefs

Cf. A000244, A000215, A007404, A051158, A051179, A169594, A356241.

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

Sequence in context: A115604 A128617 A116488 * A216601 A283000 A145765

Adjacent sequences: A356239 A356240 A356241 * A356243 A356244 A356245

Keyword

nonn

Author

Amiram Eldar, Jul 30 2022

Status

approved