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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
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
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
Subsequence of A356877.
Sequence in context: A124951 A126410 A356877 * A071634 A234252 A101804
KEYWORD
base,nonn
AUTHOR
EXTENSIONS
a(16)-a(21) from Giovanni Resta, Aug 07 2015
a(22)-a(24) from Karl-Heinz Hofmann, Oct 14 2022
STATUS
approved