A260986 Numbers n such that H(n)/H(n^2) is a new record, where H(n) = A000120(n) is the sum of the binary digits of n.

1, 23, 111, 479, 1471, 6015, 24319, 28415, 490495, 6025215, 8122367, 98549759, 132104191, 1593769983, 1862205439, 29930291199, 479961546751, 514321285119, 8237743079423, 131872659079167, 136270705590271, 35461448750596095, 7998111458938322943, 9151032963545169919

Offset

1,2

Comments

This sequence is infinite, a result which follows from Stolarsky's Theorem 2.

a(22) > 2.4*10^13. - Giovanni Resta, Aug 07 2015

a(25) > 5.8*10^20. - Karl-Heinz Hofmann, Oct 14 2022

Links

Table of n, a(n) for n=1..24.

Kevin G. Hare, Shanta Laishram, and Thomas Stoll, Stolarsky's conjecture and the sum of digits of polynomial values, Proc. Amer. Math. Soc. 139:1 (2011), pp. 39-49.

K. B. Stolarsky, The binary digits of a power, Proc. Amer. Math. Soc. 71 (1978), pp. 1-5.

Example

23 is 10111 in binary and 23^2 = 529 is 1000010001 in binary. Each smaller number has H(n)/H(n^2) <= 1, but H(23)/H(529) = 4/3 > 1, so 23 is in this sequence.

Mathematica

DeleteDuplicates[Table[{n, Total[IntegerDigits[n, 2]]/Total[IntegerDigits[n^2, 2]]}, {n, 500000}], GreaterEqual[ #1[[2]], #2[[2]]]&][[;; , 1]] (* The program generates the first 9 terms of the sequence. *) (* Harvey P. Dale, Sep 21 2023 *)

Prog

(PARI) r=2; forstep(n=1, 1e9, 2, t=hammingweight(n^2)/hammingweight(n); if(t<r, r=t; print1(n", ")))

Crossrefs

Cf. A000120, A159918, A230097.

Subsequence of A356877.

Sequence in context: A124951 A126410 A356877 * A071634 A234252 A101804

Adjacent sequences: A260983 A260984 A260985 * A260987 A260988 A260989

Keyword

base,nonn

Author

Charles R Greathouse IV, Aug 06 2015

Extensions

a(16)-a(21) from Giovanni Resta, Aug 07 2015

a(22)-a(24) from Karl-Heinz Hofmann, Oct 14 2022

Status

approved