A309714 The smallest possible nonnegative difference between the sum of the first n positive integers (A000217) and the sum of any number of the directly following and consecutive integers.

1, 0, 2, 5, 2, 6, 1, 6, 12, 5, 12, 3, 11, 0, 9, 19, 6, 17, 2, 14, 27, 10, 24, 5, 20, 36, 15, 32, 9, 27, 2, 21, 41, 14, 35, 6, 28, 51, 20, 44, 11, 36, 1, 27, 54, 17, 45, 6, 35, 65, 24, 55, 12, 44, 77, 32, 66, 19, 54, 5, 41, 78, 27, 65, 12, 51, 91, 36, 77, 20, 62

Offset: 1

Author

Bob Andriesse, Aug 13 2019

Keywords

• nonn

Comments

a(n) = 0 if a positive integer m exists, such that m * (m + 1) = 2 * n * (n + 1). Let k = m - n, then n = (2 * k - 1 + sqrt(8 * k^2 + 1)) / 2. All k for which 8 * k^2 + 1 is a perfect square (A001109) yield a value for n for which a(n) = 0.

a(A053141(n)) = 0 for all n.

Examples

			a(2) = 1 + 2 - 3 = 0;
a(3) = 1 + 2 + 3 - 4 = 2;
a(7) = 1 + 2 + 3 + 4 + 5 + 6 + 7 - (8 + 9 + 10) = 1.
a(A053141(2)) = a(14) = 0, because A000217(20) = 2 * A000217(14).
		

Crossrefs

Cf. A000217, A001109, A001652, A053141, A309655.

Programs

PARI

a(n) = {my(t=n*(n+1)/2, k = n+1); while(t >= k, t -= k; k++); t;} \\ Michel Marcus, Aug 16 2019