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%I #16 Dec 31 2022 15:27:42
%S 13,17,37,97,457,557,1117,1217,1297,2237,2377,2897,4937,7237,9277,
%T 10457,18797,21317,23557,24077,27817,29437,30757,34757,38917,39157,
%U 48157,48817,50497,55897,60617,62297,64997,72617,81157,82457,90017,94597,107837,108877,111857
%N Lesser p of a sexy prime pair such that (p-3)/2 is also the lesser prime of a sexy prime pair.
%C Equivalently, sums of the form (sexy primes - 3) which are also the lesser prime of a sexy prime pair.
%C Also numbers m such that m-4, m-1, m+5 and m+8 cannot be represented as x*y + x + y, with x >= y > 1 (A254636).
%C More generally, any sequence of numbers m such that A254636(m - 2*k - 2), A254636(m - 1), A254636(m + 4*k + 1) and A254636(m + 6*k + 2) are all 0 will only provide prime numbers which are lesser of a pair of primes (p, q) such that the pair (r, s) forms also a pair of primes, where q = p + 2*(2*k + 1), r = (p - 2*k - 1)/2, and s = (q + 2*k + 1)/2. Obviously, s - r = q - p = 2*(2*k + 1).
%C For k = 0, we get sequence A256386 (starting from its 6th term).
%C For k = 1, this sequence.
%C For k = 2, sequence starts: 19, 31, 43, 79, 127, 163, 283, 547, 751, 919, ...
%C For k = 3, sequence starts: 17, 53, 113, 593, 773, 1553, 1733, 1973, 4013, ...
%C For k = 4, sequence starts: 19, 131, 431, 811, 991, 2111, 5431, 6011, 10771, ...
%C etc.
%C For n > 1, a(n) is congruent to 17 modulo 20.
%C Number of terms < 10^k: 0, 4, 6, 15, 38, 167, 934, 5091, 30229, ...
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SexyPrimes.html">Sexy Primes</a>
%e 97 is the lesser in the sexy prime pair (97, 103), and the pair of (97-3)/2 and (103+3)/2 yields another sexy prime pair: (47, 53). Hence 97 is in the sequence.
%t Select[Prime[Range[11000]], AllTrue[Join[{#+6}, (#-3)/2 + {0,6}], PrimeQ]&] (* _Amiram Eldar_, Nov 23 2022 *)
%o (PARI) isok1(p) = isprime(p) && isprime(p+6); \\ A023201
%o isok(p) = isok1(p) && isok1((p-3)/2); \\ _Michel Marcus_, Nov 23 2022
%Y Cf. A023201, A255361, A254636, A256386.
%K nonn
%O 1,1
%A _Lamine Ngom_, Nov 23 2022