A261586 Odd numbers n such that the sum of the binary digits of n equals the sum of the binary digits of n^2.

1, 3, 7, 15, 31, 63, 79, 91, 127, 157, 159, 183, 187, 255, 279, 287, 317, 319, 351, 365, 375, 379, 445, 511, 573, 575, 637, 639, 703, 735, 751, 759, 763, 815, 893, 975, 1023, 1071, 1087, 1145, 1149, 1151, 1215, 1255, 1277, 1279, 1407, 1449, 1455, 1463

Offset: 1

Author

Charles R Greathouse IV, Aug 25 2015

Keywords

• nonn
• base

Comments

A077436 consists of elements of this sequence times powers of 2.

Hare, Laishram, & Stoll show that this sequence is infinite. In particular for each k in {12, 13, 16, 17, 18, 19, 20, ...} there are infinitely many terms in this sequence with binary digit sum k.

Subsequence of A077436.

Examples

			15 = 1111_2 and 15^2 = 11100001_2, both of which have a Hamming weight (sum of binary digits) equal to 4.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• K. G. Hare, S. Laishram, and T. Stoll, The sum of digits of n and n^2, International Journal of Number Theory 7:7 (2011), pp. 1737-1752.
• Giuseppe Melfi, On simultaneous binary expansions of n and n^2, arXiv:math/0402458 [math.NT], 2004.

Crossrefs

Cf. A077436, A254066.

Programs

Magma

[n: n in [1..1500 by 2] | &+Intseq(n, 2) eq &+Intseq(n^2, 2) ]; // Vincenzo Librandi, Aug 30 2015

Mathematica

Select[Range[1, 1463, 2], Total@ IntegerDigits[#, 2] == Total@ IntegerDigits[#^2, 2] &] (* Michael De Vlieger, Aug 29 2015 *)

PARI

is(n)=n%2 && hammingweight(n)==hammingweight(n^2)