A052350 Least prime in A001359 (lesser of twin primes) such that the distance (A053319) to the next twin is 6*n.

5, 17, 41, 617, 71, 311, 2267, 521, 1877, 461, 1721, 347, 1151, 1787, 3581, 2141, 6449, 1319, 21377, 1487, 12251, 4799, 881, 23057, 659, 19541, 12377, 2381, 38747, 10529, 37361, 8627, 9041, 33827, 5879, 80231, 15359, 45821, 36107, 14627, 37991, 36527, 87251, 70997

Offset: 1

Author

Labos Elemer, Mar 07 2000

Keywords

• nonn

Comments

Smallest distance (A052380) and also smallest possible increment of twin-distances is 6.

Primes may occur between p+2 and p+6n.

The prime a(n) determines a prime quadruple: [p, p+2, p+6n, p+6n+2] and a [2, 6n-2, 2] d-pattern.

Examples

			The first 3 terms (5, 17, 41) are followed by difference patterns as it is displayed: 5 by [2, 4, 2], 17 by [2, 4+6, 2], 41 by [2, 4+6+6, 2] determining prime quadruples: (5, 7, 11, 13), (17, 19, 29, 31) or (41, 43, 59, 61), respectively.
a(10) = 461 gives the quadruple [461, 463, 521 = 461+60, 523], and between 521 and 463, 7 primes occur.
		

Links

• Norman Luhn, Table of n, a(n) for n = 1..4624 (terms 1..4500 from Martin Raab).

Crossrefs

Cf. A001359, A007530, A052380, A052381, A053319, A113274, A113275.

Cf. A052351, A052352, A052353, A052354, A052355, A052356, A052357, A052358, A052359.

Programs

Mathematica

NextLowerTwinPrim[n_] := Block[{k = n + 6}, While[ !PrimeQ[k] || !PrimeQ[k + 2], k += 6]; k];p = 5; t = Table[0, {50}]; Do[ q = NextLowerTwinPrim[p]; d = (q - p)/6; If[d < 51 && t[[d]] == 0, t[[d]] = p; Print[{d, p}]]; p = q, {n, 1500}]; t (* Robert G. Wilson v, Oct 28 2005 *)

PARI

list(len) = {my(s = vector(len), c = 0, p1 = 5, q1 = 0, q2, d); forprime(p2 = 7, , if(p2 == p1 + 2, q2 = p1; if(q1 > 0, d = (q2 - q1)/6; if(d <= len && s[d] == 0, c++; s[d] = q1; if(c == len, return(s)))); q1 = q2); p1 = p2);} \\ Amiram Eldar, Mar 04 2025

Extensions

Name corrected by Amiram Eldar, Mar 04 2025