OFFSET
1,1
COMMENTS
This includes all k > 1 such that the average of ones in the binary expansion of k is the same of the average of ones in binary expansion of k^2; these are the values in the sequence with sqrt(2)*2^j < a(k) < 2^(j+1) for some j. - Corrected by Franklin T. Adams-Watters, Aug 23 2012
Conjecture: The counting function p(n) satisfies p(n)=c n/log n + o(n/log n).
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Giuseppe Melfi, Su alcune successioni di interi, 2° Incontro Italiano di Teoria dei Numeri, Parma, 13-15 novembre 2003.
Giuseppe Melfi, On certain positive integer sequences, arXiv:math/0404555 [math.NT], 30 Apr 2004.
EXAMPLE
21 is in the sequence because 21=10101_2 (3 1's) and 441=110111001_2 (6 1's).
MAPLE
select(t -> 2*convert(convert(t, base, 2), `+`) = convert(convert(t^2, base, 2), `+`), [$1..1000]); # Robert Israel, Aug 27 2015
MATHEMATICA
f[n_] := Total@ IntegerDigits[n, 2]; Select[Range@ 360, 2 f@ # == f[#^2] &] (* Michael De Vlieger, Aug 27 2015 *)
PROG
(PARI) isok(n) = norml2(binary(n^2)) == 2*norml2(binary(n)) \\ Michel Marcus, Jun 20 2013
CROSSREFS
KEYWORD
easy,nonn,base
AUTHOR
Giuseppe Melfi, Jun 13 2003
STATUS
approved