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Equivalently, sums of the form (sexy primes - 3) which are also a the lesser prime of a sexy prime pair.

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 primes prime pair: (47, 53). Hence 97 is in the sequence.

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Lesser p of a pair of sexy primes prime pair such that (p-3)/2 is also the lesser prime of a pair of sexy primesprime pair.

Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SexyPrimes.html">Sexy Primes</a>

97 is the lesser in the sexy primes prime pair (97, 103), and the couple pair (97-3)/2 and (103+3)/2 yields another sexy primes pair: (47, 53). Hence 97 is in the sequence.

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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).

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).

For k = 2, sequence starts: 19, 31, 43, 79, 127, 163, 283, 547, 751, 919, …...

For k = 3, sequence starts: 17, 53, 113, 593, 773, 1553, 1733, 1973, 4013, …...

For k = 4, sequence starts: 19, 131, 431, 811, 991, 2111, 5431, 6011, 10771, …...

For n > 1, a(n) is congruent to 17 modulo 20.

Number of terms < 10^k: 0, 4, 6, 15, 38, 167, 934, 5091, 30229, …...

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(PARI) isok1(p) = isprime(p) && isprime(p+6); \\ A023201

isok(p) = isok1(p) && isok1((p-3)/2); \\ Michel Marcus, Nov 23 2022

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