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%I #38 Jan 11 2023 11:09:02
%S 5,17,41,617,71,311,2267,521,1877,461,1721,347,1151,1787,3581,2141,
%T 6449,1319,21377,1487,12251,4799,881,23057,659,19541,12377,2381,38747,
%U 10529,37361,8627,9041,33827,5879,80231,15359,45821,36107,14627,37991
%N Smallest primes from A001359 where the distance (A053319) to the next similar twin increases.
%C Smallest distance (A052380) and also smallest possible increment of twin-distances is 6.
%C Primes may occur between p+2 and p+6n.
%H Norman Luhn, <a href="/A052350/b052350.txt">Table of n, a(n) for n = 1..4624</a> (terms 1..4500 from Martin Raab).
%F The prime a(n) determines a prime quadruple: [p, p+2, p+6n, p+6n+2] and a [2, 6n-2, 2] d-pattern.
%e 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.
%e a(10)=461 gives the quadruple [461, 463, 521=461+60, 523], and between 521 and 463, 7 primes occur.
%t 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 *)
%Y Cf. A001359, A053319, A007530, A052380, A052381, A113274, A113275.
%K nonn
%O 1,1
%A _Labos Elemer_, Mar 07 2000