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A337436 6*a(n) + 1 is the least upper prime p of a pair of twin primes p - 2, p, for which the prime gap immediately following p achieves the size 2*A007494(n). 1

%I #6 Sep 02 2020 19:25:11

%S 1,5,23,33,322,87,325,278,495,1293,2027,4725,3468,2690,27177,14438,

%T 4245,6773,13283,24938,104283,92067,28893,60015,119362,46905,44270,

%U 106323,90713,67475,266618,207107,139708,1496910,716182,598867,439633,688518,224922,315893

%N 6*a(n) + 1 is the least upper prime p of a pair of twin primes p - 2, p, for which the prime gap immediately following p achieves the size 2*A007494(n).

%C Apart from the atypical case [3, 5, 7], prime gaps nextprime(p+1)-p following a pair of twin primes p-2, p can only have the sizes 4, 6, 10, 12, 16, 18, ..., i.e., numbers k of the form 2*(k == 0 or 2 mod 3) = 2*A007494(n).

%e a(1) = 1: The first occurrence of 3 consecutive primes [p-2, p, p+4] is at p = 6*a(1) + 1 = 7 -> [5, 7, 11],

%e a(2) = 5: consecutive primes [p-2, p, p+6] first occur at p = 6*a(2) * 1 = 31 -> [29, 31, 37],

%e a(3) = 23: consecutive primes [p-2, p, p+10] first occur at p = 6*a(3) + 1 = 139 -> [137, 139, 149].

%Y Cf. A007497, A329160, A329161, A337435.

%K nonn

%O 1,2

%A _Hugo Pfoertner_, Sep 02 2020

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