A125770 -id:A125770 - cp's OEIS frontend

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

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

nonn

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, ...,.

			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, ...

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

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]

A125771 Primes of the form j*T_k +/- 1, where T_k is the k-th triangular number greater than 9.

Original entry on oeis.org

11, 19, 29, 31, 37, 41, 43, 59, 61, 67, 71, 73, 79, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 167, 179, 181, 191, 197, 199, 211, 223, 229, 233, 239, 241, 251, 263, 269, 271, 277, 281, 293, 307, 311, 313, 331, 337, 349, 359, 379, 389, 397

Offset: 1

Jonathan Vos Post & Robert G. Wilson v, Dec 05 2006

nonn

Since all primes would eventually appear in A125765 or A125766 because (p +/-1) times 1(1+1)/2 equals (p +/- 1) let us not use the first triangular number 1.

Primes not of the form j*T_k +/- 1, where T_k is the k-th triangular number greater than 1 only produces one prime: 3. If we restrict triangular numbers greater than 5, then only two primes are found: 2 & 3.

			11 = 1*10 +1,
19 = 2*10 -1, etc.

Cf. A000217, A125765, A125766, A125767, A125768, A125769, A125770.

s = {}; Do[m = j*k*(k + 1)/2; If[ PrimeQ[m - 1], AppendTo[s, m - 1]]; If[ PrimeQ[m + 1], AppendTo[s, m + 1]], {j, 40}, {k, 4, 23}]; Take[ Union@s, 75]

cp's OEIS Frontend

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

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