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

1,21

Comments

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

Links

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

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

Example

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.

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

Prog

(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

Crossrefs

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

Coincides with A331410 on A054784.

Sequence in context: A161371 A327928 A364387 * A091970 A093955 A330168

Adjacent sequences: A147642 A147643 A147644 * A147646 A147647 A147648

Keyword

easy,nonn

Author

Omar E. Pol, Nov 09 2008

Status

approved