A332761 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.

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, 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, 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

Offset: 0

Author

Juri-Stepan Gerasimov, Feb 23 2020

Keywords

• nonn

Comments

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

Examples

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

Crossrefs

Cf. A000290, A072273, A060594, A332802.

Programs

Magma

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

Formula

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.

a(n) = A072273(n-1). - Jinyuan Wang, Feb 25 2020

Extensions

a(70) corrected by Jinyuan Wang, Feb 25 2020