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A261593 Odd numbers n such that the sum of the binary digits of n and n^2 both equal 12.

%I A261593 #21 Aug 27 2015 18:23:51
%S A261593 4095,10239,11263,12159,12223,12255,12271,12279,12283,14333,15351,
%T A261593 15355,15743,15807,18431,19455,19967,20351,20477,22015,22495,22511,
%U A261593 24031,24303,24431,24445,25599,26615,26621,27519,27631,27639,28095,28411,28413,28511,28541,28575
%N A261593 Odd numbers n such that the sum of the binary digits of n and n^2 both equal 12.
%C A261593 Hare, Laishram, & Stoll show that this sequence is infinite.
%C A261593 12 is the least constant for which the associated sequence is known to be infinite. 1 through 8 make finite sequences, the sequences with 9 and 10 are conjectured finite, and the sequence with 11 is unknown. (See Hare-Laishram-Stoll Theorem 1.3 and 1.4 plus section 5.)
%C A261593 Odd numbers n such that the sum of the binary digits of n-1 and n^2-1 both equal 11. - _Chai Wah Wu_, Aug 26 2015
%H A261593 Charles R Greathouse IV, <a href="/A261593/b261593.txt">Table of n, a(n) for n = 1..1329</a>
%H A261593 K. G. Hare, S. Laishram, and T. Stoll, <a href="http://arxiv.org/abs/1001.4170">The sum of digits of n and n^2</a>, International Journal of Number Theory 7:7 (2011), pp. 1737-1752.
%e A261593 4095 = 111111111111_2 and 4095^2 = 111111111110000000000001_2 both have 12 1s in binary.
%t A261593 Select[2 Range@ 15000 - 1, Total@ IntegerDigits[#, 2] == 12 && Total@ IntegerDigits[#^2, 2] == 12 &] (* _Michael De Vlieger_, Aug 27 2015 *)
%o A261593 (PARI) is(n)=n%2 && hammingweight(n)==12 && hammingweight(n^2)==12
%o A261593 (PARI) \\ List the elements below 2^(N+1).
%o A261593 go(N)=my(v=List(),n); forvec(u=vector(11,i,[1,N-11+i]), n=sum(i=1,11,2^u[i])+1; if(hammingweight(n^2)==12, listput(v,n)), 2); Set(v)
%o A261593 (Python)
%o A261593 from itertools import combinations
%o A261593 A261593_list = []
%o A261593 for c in combinations((2**x for x in range(15)),11):
%o A261593     n = sum(c)
%o A261593     if sum(int(d) for d in format(n*(n+1),'b')) == 11:
%o A261593         A261593_list.append(2*n+1)
%o A261593 A261593_list = sorted(A261593_list) # _Chai Wah Wu_, Aug 26 2015
%Y A261593 Subsequence of A261586 and hence of A077436.
%K A261593 nonn,base
%O A261593 1,1
%A A261593 _Charles R Greathouse IV_, Aug 25 2015