

A358573


a(n) = smallest prime p such that q, r and s are all prime, where q = p + 2*(2*n + 1), r = (p  2*n  1)/2, and s = (q + 2*n + 1)/2.


0



11, 13, 19, 17, 19, 229, 47, 29, 163, 29, 31, 37, 47, 53, 1231, 41, 43, 61, 83, 61, 439, 1217, 59, 73, 59, 61, 67, 89, 83, 541, 71, 73, 103, 593, 271, 349, 83, 89, 103, 461, 239, 97, 107, 97, 211, 149, 107, 229, 263, 181, 499, 317, 139, 1453, 131, 809, 127, 137, 163
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OFFSET

0,1


COMMENTS

Equivalently, smallest prime of the form (p + q  2*n  1), where p is prime, q = p + 2*(2*n + 1) is prime, and (p + q + 2*n + 1) is also prime.
a(n) is the first term of the sequence of numbers m such that (m  2*n  2), (m  1), (m + 4*n + 1) and (m + 6*n + 2) cannot be represented as x*y + x + y, with x >= y > 1 (A254636).
Such sequence contains only prime numbers which are the lesser of a pair of primes (p, q) such that the pair (r, s) also forms a pair of primes with the same difference, where q = p + 2*(2*n + 1), r = (p  2*n  1)/2, and s = (q + 2*n + 1)/2.


LINKS



EXAMPLE

229 is the lesser prime in the pair (229, 251) with difference 2*(2*5+1) = 22, and the couple (22922/2)/2 = 109 and (251+22/2)/2 = 131 forms another prime pair with distance 22, and there is no prime lower than 229 with this property. Hence a(5) = 229.


MATHEMATICA

a[n_] := Module[{p=2, q, r, s}, While[!AllTrue[{(q = p + 2*(2*n + 1)), (r = (p  2*n  1)/2), (s = (q + 2*n + 1)/2)}, #>0 && PrimeQ[#] &], p = NextPrime[p]]; p]; Array[a, 60, 0] (* Amiram Eldar, Nov 23 2022 *)


PROG

(PARI) a(n) = my(p=2, q); while(!isprime(q = p + 2*(2*n + 1))  !isprime((p  2*n  1)/2)  !isprime((q + 2*n + 1)/2), p=nextprime(p+1)); p; \\ Michel Marcus, Nov 23 2022


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



