A099249 -id:A099249 - cp's OEIS frontend

A099244 Greatest common divisor of length of n in binary representation and its number of ones.

1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2, 2, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 1, 2, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Oct 08 2004

For k >= 2, n in the range [2^(k-1)..2^k - 2] have binary length k but fewer than k 1's, thus a(n) is a proper divisor of k, and if k is a prime then a(n) = 1. - Ctibor O. Zizka, Jun 19 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n.

Cf. A000120, A000225, A007088, A070939, A099245, A099246, A099247, A099248, A099249.

a099244 n = gcd (a070939 n) (a000120 n)
-- Reinhard Zumkeller, Oct 10 2013

a[n_] := GCD[BitLength[n], DigitCount[n, 2, 1]]; Array[a, 100] (* Amiram Eldar, Jul 16 2023 *)

a(n) = {my(b = binary(n)); gcd(#b, vecsum(b));} \\ Amiram Eldar, Jul 26 2025

from math import gcd
def a(n): b = bin(n)[2:]; return gcd(len(b), b.count('1'))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jun 17 2021

a(n) = gcd(A070939(n), A000120(n)).

a(A000225(n)) = n and a(m) < n for m < A000225(n).

A099247 Numbers such that, in binary representation, the length and the number of ones are coprime.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 11, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 47, 55, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103

Offset: 1

Reinhard Zumkeller, Oct 08 2004

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n.

Complement of A099248.

Cf. A070939, A000120, A007088, A099244, A099249.

a099247 n = a099247_list !! (n-1)
a099247_list = filter ((== 1) . a099244) [1..]
-- Reinhard Zumkeller, Oct 10 2013

Select[Range[150],CoprimeQ[IntegerLength[#,2],DigitCount[#,2,1]]&] (* Harvey P. Dale, Sep 22 2012 *)

isok(k) = {my(b = binary(k)); gcd(#b, vecsum(b)) == 1;} \\ Amiram Eldar, Jul 26 2025

A099244(a(n)) = 1.

Definition clarified by Harvey P. Dale, Sep 22 2012

cp's OEIS Frontend

A099244 Greatest common divisor of length of n in binary representation and its number of ones.

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