A305436 -id:A305436 - 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

A305795 Restricted growth sequence transform of A305794, a filter sequence constructed from the binary expansions of the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 13, 14, 15, 5, 16, 11, 17, 18, 19, 20, 21, 22, 19, 23, 24, 20, 25, 26, 27, 28, 10, 29, 30, 31, 19, 32, 33, 31, 34, 35, 36, 37, 38, 39, 40, 41, 42, 23, 36, 35, 43, 44, 45, 32, 38, 46, 47, 39, 48, 49, 50, 51, 52, 11, 17, 53, 54, 20, 55, 31, 56, 57, 36, 58, 59, 39, 60, 61, 56, 35, 62, 63, 64, 65, 66, 35, 67

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, A305794.

Cf. also A304105, A305793, A305815.

\\ 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; };
A305794(n) = { my(m=1); fordiv(n, d, if(d>1, m *= prime(A286622(d)-1))); (m); };
v305795 = rgs_transform(vector(up_to, n, A305794(n)));
A305795(n) = v305795[n];

For all i, j:
  a(i) = a(j) => A000005(i) = A000005(j).
  a(i) = a(j) => A007814(i) = A007814(j).
  a(i) = a(j) => A093653(i) = A093653(j).
  a(i) = a(j) => A154402(i) = A154402(j).
  a(i) = a(j) => A305436(i) = A305436(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).

A292315 Positive integers not divisible by any number of the form 2^n + 1 for n >= 0.

Original entry on oeis.org

1, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 203, 209, 211, 217, 223, 227, 229, 233, 239, 241, 247, 251, 253, 259, 263, 269, 271, 277, 281, 283, 287

Offset: 1

Jeffrey Shallit, Sep 14 2017

nonn

This is the same as odd numbers not divisible by numbers of the form 2^(2^i) + 1, i >= 0.
Asymptotically, the number of such numbers <= x is x/4 + o(x).
Composite terms are 49, 77, 91, 121, 133, 143, 161, ... - Altug Alkan, Sep 14 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Positions of zeros in A305436.

Position[Table[DivisorSum[n, 1 &, IntegerQ@ Log2[# - 1] &], {n, 288}], 0][[All, 1]] (* Michael De Vlieger, Jun 11 2018 *)

list(lim)=my(v=List(),u=[],t); lim\=1; forstep(n=1,lim,[4,2], if(gcd(n,1431655765)==1, listput(v,n))); v=Vec(v); for(i=5,logint(logint(lim-1,2),2), t=2^2^i+1; u=concat(u,t*[1..lim\t])); u=Set(u); setminus(v,u) \\ Charles R Greathouse IV, Sep 14 2017

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

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A305795 Restricted growth sequence transform of A305794, a filter sequence constructed from the binary expansions of the divisors of n.

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

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A292315 Positive integers not divisible by any number of the form 2^n + 1 for n >= 0.

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PARI