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

4095, 10239, 11263, 12159, 12223, 12255, 12271, 12279, 12283, 14333, 15351, 15355, 15743, 15807, 18431, 19455, 19967, 20351, 20477, 22015, 22495, 22511, 24031, 24303, 24431, 24445, 25599, 26615, 26621, 27519, 27631, 27639, 28095, 28411, 28413, 28511, 28541, 28575

Offset

1,1

Comments

Hare, Laishram, & Stoll show that this sequence is infinite.

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.)

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..1329

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.

Example

4095 = 111111111111_2 and 4095^2 = 111111111110000000000001_2 both have 12 1s in binary.

Mathematica

Select[2 Range@ 15000 - 1, Total@ IntegerDigits[#, 2] == 12 && Total@ IntegerDigits[#^2, 2] == 12 &] (* Michael De Vlieger, Aug 27 2015 *)

Prog

(PARI) is(n)=n%2 && hammingweight(n)==12 && hammingweight(n^2)==12
(PARI) \\ List the elements below 2^(N+1).
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)
(Python)
from itertools import combinations
A261593_list = []
for c in combinations((2**x for x in range(15)), 11):
    n = sum(c)
    if sum(int(d) for d in format(n*(n+1), 'b')) == 11:
        A261593_list.append(2*n+1)
A261593_list = sorted(A261593_list) # Chai Wah Wu, Aug 26 2015

Crossrefs

Subsequence of A261586 and hence of A077436.

Sequence in context: A038463 A359084 A043452 * A290869 A075952 A075957

Adjacent sequences: A261590 A261591 A261592 * A261594 A261595 A261596

Keyword

nonn,base

Author

Charles R Greathouse IV, Aug 25 2015

Status

approved