A147645 Number of distinct Mersenne primes dividing n.

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

Offset: 1

Author

Omar E. Pol, Nov 09 2008

Keywords

• easy
• nonn

Comments

a(n) = m first occurs at n = A098918(m). - Robert Israel, Feb 03 2020

Examples

			a(21)=2 because 1, 3, 7 and 21 are divisors of 21. Then 21 has two divisors that are Mersenne primes (A000668): 3 and 7.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..131072 (terms 1..10000 from Robert Israel)

Crossrefs

Cf. A000265, A000668, A001221, A080225, A098918, A154402, A173898, A353786.

Coincides with A331410 on A054784.

Programs

Maple

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for i from 1 do
m:= numtheory:-mersenne([i]);
if m > N then break fi;
for j from m by m to N do
    V[j]:= V[j]+1
od od:
convert(V,list); # Robert Israel, Feb 03 2020

PARI

A147645(n) = { my(m=3,s=0); while(m<=n, s += (isprime(m)*!(n%m)); m += (m+1)); (s); }; \\ Antti Karttunen, May 12 2022

Formula

From Antti Karttunen, May 12 2022: (Start)

a(n) = A154402(n) - A353786(n)

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

a(n) <= A331410(n). (End)

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A173898 = 0.516454... . - Amiram Eldar, Dec 31 2023