A379676 For n >= 0, a(n) is the least k >= 2 such that (n + 1)*(2*k + n) / 2 is a triangular number (A000217).

3, 7, 4, 15, 7, 5, 10, 31, 13, 6, 16, 12, 19, 10, 7, 63, 25, 11, 28, 22, 8, 15, 34, 21, 37, 16, 40, 9, 43, 20, 46, 127, 14, 21, 18, 10, 55, 25, 15, 19, 61, 26, 64, 45, 11, 30, 70, 44, 73, 31, 21, 55, 79, 35, 12, 70, 22, 36, 88, 18, 91, 40, 34, 255, 31, 13, 100, 19, 28, 24, 106, 92, 109, 46, 29, 78, 25, 14, 118, 91, 121, 51, 124, 63, 42, 55, 35, 39, 133, 43, 15

Offset: 0

Author

Ctibor O. Zizka, Dec 29 2024

Keywords

• nonn

Comments

Also for n >= 0, a(n) is the least k >= 2 such that the Sum_{i = 0..n} (k + i) is a triangular number (A000217). For k = 0, 1 the Sum is a triangular number for all n. The sequences A076114 and A076116 are for square sum and cube sum.

Examples

			n = 4: the least k >= 2 such that (4 + 1)*(2*k + 4)/2 = 5*k + 10 is a triangular number is k = 7, thus a(4) = 7.
n = 5: the least k >= 2 such that (5 + 1)*(2*k + 5)/2 = 6*k + 15 is a triangular number is k = 5, thus a(5) = 5.
		

Crossrefs

Cf. A000217, A076114, A076116, A083390.

Programs

Mathematica

a[n_] := Module[{k = 2}, While[! IntegerQ[Sqrt[4*(n + 1)*(2*k + n) + 1]], k++]; k]; Array[a, 100, 0] (* Amiram Eldar, Dec 30 2024 *)

PARI

a(n) = my(k=2); while (!ispolygonal((n + 1)*(2*k + n)/2, 3), k++); k; \\ Michel Marcus, Dec 30 2024

Formula

For i >= 0, a(2^i - 1) = 2^(i + 2) - 1, max. values of a(n).

For i >= 0, a(i*(i + 3)/2) = i + 3, min. values of a(n).

For i >= 1, i is not from A083390, a(2*i) = (3*i + 1).