A372604 The maximal exponent in the prime factorization of the largest divisor of n whose number of divisors is a power of 2.

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset

1,8

Comments

First differs from A331273 at n = 32.

Differs from A368247 at n = 1, 128, 216, 256, 384, 432, 512, ... .

All the terms are of the form 2^k-1 (A000225).

Links

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

Formula

a(n) = A051903(A372379(n)).

a(n) = A092323(A051903(n)+1).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{i>=1} 2^i * (1 - 1/zeta(2^(i+1)-1)) = 1.36955053734097783559... .

Example

4 has 3 divisors, 1, 2 and 4. The number of divisors of 4 is 3, which is not a power of 2. The number of divisors of 2 is 2, which is a power of 2. Therefore, A372379(4) = 2 and a(4) = A051903(2) = 1.

Mathematica

f[n_] := 2^Floor[Log2[n + 1]] - 1; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

Prog

(PARI) s(n) = 2^exponent(n+1) - 1;
a(n) = if(n>1, s(vecmax(factor(n)[, 2])), 0);

Crossrefs

Cf. A000225, A051903, A092323, A372379, A372466.

Cf. A331273, A368247.

Similar sequences: A007424, A368781, A372601, A372602, A372603.

Sequence in context: A031227 A185358 A368247 * A331273 A363332 A372601

Adjacent sequences: A372601 A372602 A372603 * A372605 A372606 A372607

Keyword

nonn,easy

Author

Amiram Eldar, May 07 2024

Status

approved