A353786 -id:A353786 - cp's OEIS frontend

A154402 Inverse Moebius transform of Fredholm-Rueppel sequence, cf. A036987.

1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 2, 1, 3, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 4, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 3, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 4

Offset: 1

Vladeta Jovovic, Jan 08 2009

Number of ways to write n as a sum a_1 + ... + a_k where the a_i are positive integers and a_i = 2 * a_{i-1}, cf. A000929.

Number of divisors of n of the form 2^k - 1 (A000225) for k >= 1. - Jeffrey Shallit, Jan 23 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 10000 terms from Robert Israel)

Cf. A000225, A000929, A001511, A036987, A065442, A147645, A161790 (positions of 1's), A305426, A353786.

Cf. also A305436.

N:= 200: # to get a(1)..a(N)
A:= Vector(N):
for k from 1 do
   t:= 2^k-1;
   if t > N then break fi;
   R:= [seq(i,i=t..N,t)];
   A[R]:= map(`+`,A[R],1)
od:
convert(A,list); # Robert Israel, Jan 23 2017

Table[DivisorSum[n, 1 &, IntegerQ@ Log2[# + 1] &], {n, 105}] (* Michael De Vlieger, Jun 11 2018 *)

A209229(n) = (n && !bitand(n,n-1));
A036987(n) = A209229(1+n);
A154402(n) = sumdiv(n,d,A036987(d)); \\ Antti Karttunen, Jun 11 2018

A154402(n) = { my(m=1,s=0); while(m<=n, s += !(n%m); m += (m+1)); (s); }; \\ Antti Karttunen, May 12 2022

G.f.: Sum_{k>0} x^(2^k-1)/(1-x^(2^k-1)).
From Antti Karttunen, Jun 11 2018: (Start)
a(n) = Sum_{d|n} A036987(d).
a(n) = A305426(n) + A036987(n). (End)
a(n) = A147645(n) + A353786(n). - Antti Karttunen, May 12 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A065442 = 1.606695... . - Amiram Eldar, Dec 31 2023

A147645 Number of distinct Mersenne primes dividing n.

Original entry on oeis.org

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

Omar E. Pol, Nov 09 2008

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

			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.

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

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

Coincides with A331410 on A054784.

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

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

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

cp's OEIS Frontend

A154402 Inverse Moebius transform of Fredholm-Rueppel sequence, cf. A036987.

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A147645 Number of distinct Mersenne primes dividing n.

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