A357219 -id:A357219 - cp's OEIS frontend

A357218 Primes p such that T(p) - 2 is prime, where T(p) is the triangular number (A000217) with index p.

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 149, 157, 193, 197, 233, 257, 269, 277, 281, 313, 337, 389, 401, 409, 457, 509, 521, 541, 613, 641, 673, 701, 797, 857, 877, 881, 929, 953, 997, 1009, 1033, 1093, 1109, 1117, 1129, 1153, 1193, 1297, 1301, 1373, 1381, 1433, 1481, 1493

Offset: 1

Bernard Schott, Sep 18 2022

nonn

T(p) must be odd, so these primes p satisfy p == 1 (mod 4) (A002144).
Corresponding values of T(p)-2 are in A357219.
The first eleven primes == 1 (mod 4) are terms. The smallest Pythagorean prime that is not a term is A002144(12) = 101 because T(101) = 5151 and 5151 - 2 = 5149 = 19 * 271 (see Wells reference).

			T(5) - 2 = 5*6/2 - 2 = 13, hence 5 is a term.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 496, page 142.

Cf. A000217, A231847, A357219.

Subsequence of A002144.

filter := p -> isprime(p) and irem(p-1, 4) = 0 and isprime(p*(p+1)/2 -2) : select(filter, [$1 .. 1500]);

Select[Prime[Range[240]], PrimeQ[#*(# + 1)/2 - 2] &] (* Amiram Eldar, Sep 18 2022 *)

cp's OEIS Frontend

A357218 Primes p such that T(p) - 2 is prime, where T(p) is the triangular number (A000217) with index p.

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