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

Author

Amiram Eldar, May 07 2024

Keywords

• nonn
• easy

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).

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A331273, A368247.

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

Programs

Mathematica

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

PARI

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

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... .