A077436 Let B(n) be the sum of binary digits of n. This sequence contains n such that B(n) = B(n^2).

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 28, 30, 31, 32, 48, 56, 60, 62, 63, 64, 79, 91, 96, 112, 120, 124, 126, 127, 128, 157, 158, 159, 182, 183, 187, 192, 224, 240, 248, 252, 254, 255, 256, 279, 287, 314, 316, 317, 318, 319, 351, 364, 365, 366, 374, 375, 379, 384

Offset: 1

Author

Giuseppe Melfi, Nov 30 2002

Keywords

• easy
• nonn
• base

Comments

Superset of A023758.

Hare, Laishram, & Stoll show that this sequence contains infinitely many odd numbers. 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. - Charles R Greathouse IV, Aug 25 2015

Examples

			The element 79 belongs to the sequence because 79=(1001111) and 79^2=(1100001100001), so B(79)=B(79^2)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10476, all terms <= 2^20
• Karam Aloui, Damien Jamet, Hajime Kaneko, Steffen Kopecki, Pierre Popoli, and Thomas Stoll, On the binary digits of n and n^2, arXiv:2203.05451 [math.NT], 2022.
• 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.
• Giuseppe Melfi, Su alcune successioni di interi (English with an Italian title)

Crossrefs

Cf. A058369, A000120, A000290, A083567.

Cf. A211676 (number of n-bit numbers in this sequence).

A261586 is a subsequence. Subsequence of A352084.

Programs

Haskell

import Data.List (elemIndices)
import Data.Function (on)
a077436 n = a077436_list !! (n-1)
a077436_list = elemIndices 0
   $ zipWith ((-) `on` a000120) [0..] a000290_list
-- Reinhard Zumkeller, Apr 12 2011

Magma

[n: n in [0..400] | &+Intseq(n, 2) eq &+Intseq(n^2, 2)]; // Vincenzo Librandi, Aug 30 2015

Maple

select(t -> convert(convert(t,base,2),`+`) = convert(convert(t^2,base,2),`+`), [$0..1000]); # Robert Israel, Aug 27 2015

Mathematica

t={}; Do[If[DigitCount[n, 2, 1] == DigitCount[n^2, 2, 1], AppendTo[t, n]], {n, 0, 364}]; t
f[n_] := Total@ IntegerDigits[n, 2]; Select[Range[0, 384], f@ # == f[#^2] &] (* Michael De Vlieger, Aug 27 2015 *)

PARI

is(n)=hammingweight(n)==hammingweight(n^2) \\ Charles R Greathouse IV, Aug 25 2015

Python

def ok(n): return bin(n).count('1') == bin(n**2).count('1')
print([m for m in range(400) if ok(m)]) # Michael S. Branicky, Mar 11 2022

Formula

A159918(a(n)) = A000120(a(n)). - Reinhard Zumkeller, Apr 25 2009

Extensions

Initial 0 added by Reinhard Zumkeller, Apr 28 2012, Apr 12 2011