A305426 -id:A305426 - 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

A305793 Restricted growth sequence transform of A305792, a filter sequence constructed from binary expansions of the proper divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 15, 10, 18, 2, 19, 2, 20, 21, 7, 22, 23, 2, 15, 21, 24, 2, 25, 2, 26, 27, 28, 2, 29, 30, 31, 10, 26, 2, 32, 33, 34, 21, 28, 2, 35, 2, 36, 37, 38, 33, 39, 2, 13, 40, 41, 2, 42, 2, 43, 44, 26, 45, 46, 2, 47, 48, 43, 2, 49, 50, 51, 40, 52, 2, 53, 45, 54, 55, 56, 33, 57, 2, 58, 59

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A278222, A286622, A305792, A305795.

Cf. also A293215, A300833, A304103, A305813.

\\ Needs also code from A286622:
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A305792(n) = { my(m=1); fordiv(n,d,if(dA286622(d)-1))); (m); };
v305793 = rgs_transform(vector(up_to, n, A305792(n)));
A305793(n) = v305793[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A292257(i) = A292257(j).
  a(i) = a(j) => A305426(i) = A305426(j).
  a(i) = a(j) => A305435(i) = A305435(j).

A305435 Number of proper divisors of n of the form 2^k + 1 for k >= 0.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 2, 0, 1, 2, 1, 0, 3, 0, 2, 1, 1, 0, 2, 1, 1, 2, 1, 0, 3, 0, 1, 1, 2, 1, 3, 0, 1, 1, 2, 0, 2, 0, 1, 3, 1, 0, 2, 0, 2, 2, 1, 0, 3, 1, 1, 1, 1, 0, 3, 0, 1, 2, 1, 1, 3, 0, 2, 1, 2, 0, 3, 0, 1, 2, 1, 0, 2, 0, 2, 2, 1, 0, 2, 2, 1, 1, 1, 0, 4, 0, 1, 1, 1, 1, 2, 0, 1, 3, 2, 0, 3, 0, 1, 2

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

a(n) is the number of terms of A000051 less than n that divide n.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000051, A209229, A305436.

Cf. also A305426.

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

A209229(n) = (n && !bitand(n,n-1));
A305435(n) = sumdiv(n,d,(dA209229(d-1));

a(n) = Sum_{d|n, dA209229(d-1).

a(n) = A305436(n) - A209229(n-1).

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

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A305793 Restricted growth sequence transform of A305792, a filter sequence constructed from binary expansions of the proper divisors of n.

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PARI

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A305435 Number of proper divisors of n of the form 2^k + 1 for k >= 0.

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Mathematica

PARI

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