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Primes p such that T(p) - 2 is prime, where T(p) is the triangular number (A000217) with index p.
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%I #28 Sep 21 2022 01:38:23

%S 5,13,17,29,37,41,53,61,73,89,97,149,157,193,197,233,257,269,277,281,

%T 313,337,389,401,409,457,509,521,541,613,641,673,701,797,857,877,881,

%U 929,953,997,1009,1033,1093,1109,1117,1129,1153,1193,1297,1301,1373,1381,1433,1481,1493

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

%C T(p) must be odd, so these primes p satisfy p == 1 (mod 4) (A002144).

%C Corresponding values of T(p)-2 are in A357219.

%C 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).

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

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

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

%t Select[Prime[Range[240]], PrimeQ[#*(# + 1)/2 - 2] &] (* _Amiram Eldar_, Sep 18 2022 *)

%Y Cf. A000217, A231847, A357219.

%Y Subsequence of A002144.

%K nonn

%O 1,1

%A _Bernard Schott_, Sep 18 2022