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