A099245 -id:A099245 - 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).

A099246 Denominator of relative frequency of number of ones in the binary representation of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 3, 1, 4, 2, 2, 4, 2, 4, 4, 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 1, 6, 3, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 3, 3, 6, 3, 2, 2, 3, 2, 3, 3, 6, 2, 3, 3, 6, 3, 6, 6, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Reinhard Zumkeller, Oct 08 2004

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n.

Cf. A000120, A000225, A007088, A070939, A099244, A099245 (numerators).

import Data.Ratio ((%), denominator)
a099246 n = denominator $ (a000120 n) % (a070939 n)
-- Reinhard Zumkeller, Oct 10 2013

a[n_] := Denominator[First[#]/Total[#]] & @ DigitCount[n, 2, {1, 0}]; Array[a, 100, 0] (* Amiram Eldar, Apr 05 2025 *)

a(n)*A000120(n) = A099245(n)*A070939(n).
a(n) = A070939(n)/A099244(n) for n > 0.
a(n) = 1 iff n = A000225(k).
a(n) = if n=0 then 1 else A070939(n)/GCD(A070939(n), A000120(n)).

A293198 a(n) is the least positive k such that f(k) = f(k + n) where f(k) = A000120(k) / A070939(k).

Original entry on oeis.org

1, 5, 1, 9, 3, 21, 1, 2, 2, 19, 2, 38, 3, 37, 1, 33, 15, 35, 38, 37, 84, 35, 76, 12, 7, 10, 9, 10, 3, 4, 1, 10, 4, 2, 5, 2, 2, 6, 5, 2, 2, 5, 2, 9, 4, 6, 5, 2, 2, 5, 2, 6, 5, 5, 2, 5, 7, 137, 138, 134, 3, 133, 1, 129, 63, 131, 134, 133, 140, 131, 138, 137, 152, 139, 134, 133, 148, 131, 146, 56, 336, 135, 150, 52

Offset: 0

Altug Alkan, Oct 05 2017

Numbers m such that a(2^m*(2^(m + 1) - 1) + 1) = 2^m are 0, 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, ...
Numbers t such that a(t) = 2 are 7, 8, 10, 33, 35, 36, 39, 40, 42, 47, 48, 50, ...
Numbers t such that a(t) > t are 0, 1, 3, 5, 9, 11, 13, 15, 17, 18, 19, 20, 21, ...

			a(5) = 21 because 21 = 2^4 + 2^2 + 2^0, 21 + 5 = 2^4 + 2^3 + 2^1; A000120(21) / A070939(21) = A000120(21 + 5) / A070939(21 + 5) and 21 is the least number with this property.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Altug Alkan, Line graph of a(n+1)-a(n) for n <= 2^13
Altug Alkan, A logarithmic scatterplot of a(n)

Cf. A000120, A070939, A099245, A099246, A292895.

a(n) = {my(k=1); while (hammingweight(k+n)/#binary(k+n) != hammingweight(k) /#binary(k), k++); k;}

a(n) <> n for all n >= 0.
a(n) <= 5*n for all n >= 1.
a(2^m - 1) = 1 for all m >= 1.
a(2^m - 2^2) = 2^2 - 1 for all m >= 3.
a(2^m - 2^3) = 2^3 - 1 for all m >= 5.
a(2^m - 2^4) = 2^4 - 1 for all m >= 7.
a(2^m - 2^5) = 2^5 - 1 for all m >= 10.
a(2^m - 2^6) = 2^6 - 1 for all m >= 13.
a(2^m - 2^7) = 2^7 - 1 for all m >= 17.
a(2^m - 2^8) = 2^8 - 1 for all m >= 21.
a(2^m - 2^9) = 2^9 - 1 for all m >= 26.
a(2^(p - 1)) = 2^(p - 1) - 1 and a(2^(p - 1) - 1) = 2^p + 1 for all primes p.
a(2^(p - 1) + 1) = 2^p + 3 for all primes p >= 5.

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

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A099246 Denominator of relative frequency of number of ones in the binary representation of n.

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A293198 a(n) is the least positive k such that f(k) = f(k + n) where f(k) = A000120(k) / A070939(k).

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