A353786 Number of distinct nonprime numbers of the form 2^k - 1 that divide n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset

1,15

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Formula

a(n) = A154402(n) - A147645(n).

a(n) = a(2*n) = a(A000265(n)).

For all primes p, a(p) = 1.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A135972(n) = A065442 - A173898 = 1.0902409734... . - Amiram Eldar, Dec 31 2023

Example

Divisors of 255 are [1, 3, 5, 15, 17, 51, 85, 255], of these of the form 2^k - 1 (A000225) are 1, 3, 15 and 255, but only three of them are counted (because 3 is a prime), therefore a(255) = 3.

Mathematica

a[n_] := DivisorSum[n, 1 &, !PrimeQ[#] && # + 1 == 2^IntegerExponent[# + 1, 2] &]; Array[a, 120] (* Amiram Eldar, May 12 2022 *)

Prog

(PARI) A353786(n) = { my(m=1, s=0); while(m<=n, s += (!isprime(m))*!(n%m); m += (m+1)); (s); };

Crossrefs

Cf. A000225, A000265, A147645, A154402.

Cf. A065442, A135972, A173898.

Sequence in context: A062379 A325625 A043288 * A057523 A083235 A307609

Adjacent sequences: A353783 A353784 A353785 * A353787 A353788 A353789

Keyword

nonn

Author

Antti Karttunen, May 12 2022

Status

approved