cp's OEIS Frontend

A231898 a(n) = smallest k with property that for all m >= k, there is a square N^2 whose binary expansion contains exactly n 1's and m 0's; or -1 if no such k exists.

%I A231898 #32 Nov 21 2013 07:38:02
%S A231898 -1,-1,2,-1,4,3,4,3,4,5,5,5,6,5,5,5,6,6,6,6,6,6,6,6,6,6
%N A231898 a(n) = smallest k with property that for all m >= k, there is a square N^2 whose binary expansion contains exactly n 1's and m 0's; or -1 if no such k exists.
%C A231898 a(n) = -1 for n = 1, 2 and 4, because all squares with exactly 1, 2 or 4 1's in their binary expansion must contain an even number of 0's.
%C A231898 Conjecture: Apart from n=1, 2 and 4, no other a(n) is -1.
%C A231898 See A214560 for a related conjecture.
%e A231898 Here is a table whose columns give:
%e A231898 N, N^2, number of bits in N^2, number of 1's in N^2, number of 0's in N^2:
%e A231898 0 0 1 0 1
%e A231898 1 1 1 1 0
%e A231898 2 4 3 1 2
%e A231898 3 9 4 2 2
%e A231898 4 16 5 1 4
%e A231898 5 25 5 3 2
%e A231898 6 36 6 2 4
%e A231898 7 49 6 3 3
%e A231898 8 64 7 1 6
%e A231898 9 81 7 3 4
%e A231898 10 100 7 3 4
%e A231898 11 121 7 5 2
%e A231898 12 144 8 2 6
%e A231898 13 169 8 4 4
%e A231898 14 196 8 3 5
%e A231898 15 225 8 4 4
%e A231898 16 256 9 1 8
%e A231898 17 289 9 3 6
%e A231898 18 324 9 3 6
%e A231898 19 361 9 5 4
%e A231898 ...
%e A231898 a(n) is defined by the property that for all m >= a(n), the table contains a row ending n m. For example, there are rows ending 3 2, 3 3, 3 4, 3 5, ..., but not 3 1, so a(3) = 2.
%e A231898 a(5)=4: for t>=0, (11*2^t)^2 contains 5 1's and 2t+2 0's and (25*2^t)^2 contains 5 1's and 2t+5 0's, so for m >= 4 there is a number N such that N^2 contains 5 1's and m 0's. Also 4 is the smallest number with this property, so a(5) = 4.
%Y A231898 Cf. A000120, A023416, A159918, A214560, A230097, A231897.
%K A231898 sign,more
%O A231898 1,3
%A A231898 _N. J. A. Sloane_, Nov 19 2013
%E A231898 Missing word in definition supplied by _Jon Perry_, Nov 20 2013.