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

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

0,2

Comments

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, ...

Links

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)

Formula

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.

Example

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.

Prog

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

Crossrefs

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

Sequence in context: A103133 A098318 A360750 * A193029 A094650 A189234

Adjacent sequences: A293195 A293196 A293197 * A293199 A293200 A293201

Keyword

nonn,base,look,easy

Author

Altug Alkan, Oct 05 2017

Status

approved