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

1,3

Comments

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

Links

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

Formula

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

Maple

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

Mathematica

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

Prog

(PARI)
A209229(n) = (n && !bitand(n, n-1));
A036987(n) = A209229(1+n);
A154402(n) = sumdiv(n, d, A036987(d)); \\ Antti Karttunen, Jun 11 2018
(PARI) A154402(n) = { my(m=1, s=0); while(m<=n, s += !(n%m); m += (m+1)); (s); }; \\ Antti Karttunen, May 12 2022

Crossrefs

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

Cf. also A305436.

Sequence in context: A006345 A122497 A350330 * A210682 A293433 A177025

Adjacent sequences: A154399 A154400 A154401 * A154403 A154404 A154405

Keyword

easy,nonn

Author

Vladeta Jovovic, Jan 08 2009

Status

approved