A305435 -id:A305435 - cp's OEIS frontend

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

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).

A305436 Number of divisors of n of the form 2^k + 1 for k >= 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 1, 3, 0, 2, 1, 1, 0, 2, 1, 1, 2, 1, 0, 3, 0, 1, 2, 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, 2, 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 that divide n.

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

Cf. A000051, A209229, A292315 (positions of zeros), A305435, A323482.

Cf. also A154402.

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

A209229(n) = (n && !bitand(n,n-1));
A305436(n) = sumdiv(n,d,A209229(d-1));

a(n) = Sum_{d|n} A209229(d-1).
a(n) = A305435(n) + A209229(n-1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A323482 = 1.264499... . - Amiram Eldar, Dec 31 2023

A305426 Number of proper divisors of n of the form 2^k - 1 for k >= 1.

Original entry on oeis.org

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

Antti Karttunen, Jun 11 2018

nonn

a(n) is the number of terms of A000225 less than n that divide n.

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

Cf. A000225, A036987, A154402.

Cf. also A305435.

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

A209229(n) = (n && !bitand(n,n-1));
A036987(n) = A209229(1+n);
A305426(n) = sumdiv(n,d,(dA036987(d));

a(n) = Sum_{d|n, dA036987(d).

a(n) = A154402(n) - A036987(n).

cp's OEIS Frontend

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

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A305436 Number of divisors of n of the form 2^k + 1 for k >= 0.

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A305426 Number of proper divisors of n of the form 2^k - 1 for k >= 1.

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