A125769 a(n) is the least number j such that j*T_k +/- 1 is n-th prime for some k-th triangular number.

1, 2, 1, 1, 1, 2, 3, 2, 4, 1, 2, 1, 2, 2, 8, 9, 4, 4, 1, 2, 2, 1, 3, 2, 16, 10, 17, 3, 2, 4, 6, 2, 1, 5, 10, 10, 2, 27, 6, 29, 4, 2, 1, 32, 3, 3, 1, 8, 38, 23, 3, 2, 2, 7, 43, 4, 6, 2, 1, 10, 47, 14, 2, 4, 4, 53, 5, 12, 58, 35, 59, 3, 61, 62, 1, 64, 5, 6, 40, 3, 2, 2, 12, 12, 8, 74, 10, 76, 2, 2, 6, 4

Offset: 1

Author

Jonathan Vos Post & Robert G. Wilson v, Dec 01 2006

Keywords

• nonn

Comments

Eventually all primes p appear since (p +/-1) times 1(1+1)/2 equals (p +/- 1).

If we asked for the least number k then k always equals 1 since all primes p appear since (p +/-1) times 1(1+1)/2 equals (p +/- 1).

The k's for the corresponding j's are: round(sqrt(2p/j)).

First occurrence of i is A125770: 1, 2, 7, 9, 34, 31, 54, 15, 16, 26, 148, 68, 398, 62, 193, 25, 27, 140, 550, 397, 107, 113, ...,.

Examples

			a(1) = 1 because 1*1+1 = 2 which is the first prime,
a(2) = 2 because 2*1+1 = 3 which is the second prime,
a(3) = 4 because 1*6-1 = 5 which is the third prime,
a(8) = 3 because 2*10-1 = 19 which is the eighth prime, ...
		

Crossrefs

Cf. A000217, A125765, A125766, A125767, A125768, A125770.

Programs

Mathematica

triQ[n_] := IntegerQ@ Sqrt[8n + 1]; f[n_] := Block[{j = 1, p = Prime@n}, While[ !triQ[(p - 1)/j] && !triQ[(p + 1)/j], j++ ]; j]; Array[f, 92]