cp's OEIS Frontend

A147645 Number of distinct Mersenne primes dividing n.

%I A147645 #22 Dec 31 2023 06:23:41
%S A147645 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,
%T A147645 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,
%U A147645 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
%N A147645 Number of distinct Mersenne primes dividing n.
%C A147645 a(n) = m first occurs at n = A098918(m). - _Robert Israel_, Feb 03 2020
%H A147645 Antti Karttunen, <a href="/A147645/b147645.txt">Table of n, a(n) for n = 1..131072</a> (terms 1..10000 from Robert Israel)
%F A147645 From _Antti Karttunen_, May 12 2022: (Start)
%F A147645 a(n) = A154402(n) - A353786(n)
%F A147645 a(n) = a(2*n) = a(A000265(n)).
%F A147645 a(n) <= A331410(n). (End)
%F A147645 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A173898 = 0.516454... . - _Amiram Eldar_, Dec 31 2023
%e A147645 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.
%p A147645 N:= 100: # for a(1)..a(N)
%p A147645 V:= Vector(N):
%p A147645 for i from 1 do
%p A147645 m:= numtheory:-mersenne([i]);
%p A147645 if m > N then break fi;
%p A147645 for j from m by m to N do
%p A147645     V[j]:= V[j]+1
%p A147645 od od:
%p A147645 convert(V,list); # _Robert Israel_, Feb 03 2020
%o A147645 (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
%Y A147645 Cf. A000265, A000668, A001221, A080225, A098918, A154402, A173898, A353786.
%Y A147645 Coincides with A331410 on A054784.
%K A147645 easy,nonn
%O A147645 1,21
%A A147645 _Omar E. Pol_, Nov 09 2008