A358571 Lesser p of a sexy prime pair such that (p-3)/2 is also the lesser prime of a sexy prime pair.

13, 17, 37, 97, 457, 557, 1117, 1217, 1297, 2237, 2377, 2897, 4937, 7237, 9277, 10457, 18797, 21317, 23557, 24077, 27817, 29437, 30757, 34757, 38917, 39157, 48157, 48817, 50497, 55897, 60617, 62297, 64997, 72617, 81157, 82457, 90017, 94597, 107837, 108877, 111857

Offset: 1

Author

Lamine Ngom, Nov 23 2022

Keywords

• nonn

Comments

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

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 = 0, we get sequence A256386 (starting from its 6th term).

For k = 1, this sequence.

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, ...

etc.

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, ...

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Sexy Primes

Crossrefs

Cf. A023201, A255361, A254636, A256386.

Programs

Mathematica

Select[Prime[Range[11000]], AllTrue[Join[{#+6}, (#-3)/2 + {0,6}], PrimeQ]&] (* Amiram Eldar, Nov 23 2022 *)

PARI

isok1(p) = isprime(p) && isprime(p+6); \\ A023201
isok(p) = isok1(p) && isok1((p-3)/2); \\ Michel Marcus, Nov 23 2022