A332761 - OEIS

%I #30 Sep 08 2022 08:46:25

%S 0,1,0,0,1,1,1,1,1,2,1,1,1,2,1,1,2,2,1,1,1,2,2,1,1,3,1,1,1,2,1,2,1,2,

%T 2,1,2,2,1,1,2,3,1,2,1,2,2,1,1,3,1,1,2,2,1,1,2,3,2,1,1,3,1,1,2,2,2,2,

%U 1,2,2,2,1,3,1,1,2,2,2,2,1,3,1,1,1,3,2,1,2,3,1,2,2,2,2,1,2

%N Exponents m such that the number of nonnegative k <= n, possessing the property that n + n*k - k is a square, is equal to 2^m.

%C Where records occur gives 0, 1, 9, 25, 121, 841, 9241, ...

%F a(n+2) is the exponent r if 2^r is equal to the number of squares of the form k + k*n - n, 0 <= k <= n.

%F a(n) = A072273(n-1). - _Jinyuan Wang_, Feb 25 2020

%e a(0) = 0 because 0 + 0*0 - 0 = 1 = 1^2 and 1 = 2^0.

%e a(1) = 1 because 1 + 1*0 - 0 = 1 = 1^2, 1 + 1*1 - 1 = 1^2 and 2 = 2^1.

%e a(9) = 2 because 9 + 9*0 - 0 = 9 = 3^2, 9 + 9*2 - 2 = 25 = 5^2, 9 + 9*8 - 8 = 64 = 8^2, 9 + 9*9 - 9 = 81 = 9^2 and 4 = 2^2.

%o (Magma) [[m: m in [0..n] | #[k: k in [0..n] | IsSquare(n+n*k-k)] eq 2^m]: n in [0..100]];

%Y Cf. A000290, A072273, A060594, A332802.

%K nonn

%O 0,10

%A _Juri-Stepan Gerasimov_, Feb 23 2020

%E a(70) corrected by _Jinyuan Wang_, Feb 25 2020