A356877 -id:A356877 - cp's OEIS frontend

A357750 a(n) is the least k such that B(k^2) - B(k) = n, where B(m) is the binary weight A000120(m).

0, 5, 11, 21, 45, 75, 217, 331, 181, 789, 1241, 2505, 5701, 5221, 11309, 19637, 43151, 69451, 82709, 166027, 346389, 607307, 689685, 1458357, 1380917, 2507541, 5906699, 2965685, 5931189, 11862197, 47448787, 82188309, 57804981, 94905541, 188883211, 373457573, 640164021

Offset: 0

Karl-Heinz Hofmann and Hugo Pfoertner, Oct 17 2022

			  ----------------------------------------------------
  n     k      k^2     binary k             binary k^2
  ----------------------------------------------------
  0     0        0            0                      0
  1     5       25          101                  11001
  2    11      121         1011                1111001
  3    21      441        10101              110111001
  4    45     2025       101101            11111101001
  5    75     5625      1001011          1010111111001
  6   217    47089     11011001       1011011111110001
  7   331   109561    101001011      11010101111111001
  8   181    32761     10110101        111111111111001
  9   789   622521   1100010101   10010111111110111001

Cf. A000120, A000290, A159918, A164343, A164344, A356877, A357658.

a(n) = my(k=0); while(hammingweight(k^2) - hammingweight(k) != n, k++); k;

from itertools import count
def A357750(n):
    for k in count(0):
        if (k**2).bit_count()-k.bit_count()==n:
            return k # Chai Wah Wu, Oct 17 2022

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.

Original entry on oeis.org

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

Charles R Greathouse IV, Aug 06 2015

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

			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.

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.

Cf. A000120, A159918, A230097.

Subsequence of A356877.

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 *)

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

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

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

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A357750 a(n) is the least k such that B(k^2) - B(k) = n, where B(m) is the binary weight A000120(m).

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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.

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