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

1,3

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.

Links

Table of n, a(n) for n=1..71.

Example

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).

Prog

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

Crossrefs

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

Sequence in context: A198253 A159989 A195702 * A309431 A344077 A198539

Adjacent sequences: A309711 A309712 A309713 * A309715 A309716 A309717

Keyword

nonn

Author

Bob Andriesse, Aug 13 2019

Status

approved