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A141657 a(n) = (prime(2*m)-prime(m))/6 where m is the n-th positive integer such that (prime(2*m)-prime(m))/6 is prime.

%I A141657 #22 Jul 02 2023 09:09:22
%S A141657 2,3,7,11,13,17,19,23,71,71,83,83,89,101,103,127,163,167,191,193,199,
%T A141657 239,239,251,263,277,307,307,307,317,331,337,389,443,463,487,509,509,
%U A141657 557,659,691,787,797,863,881,883,887,887,953,971,977,991,1019,1063,1069
%N A141657 a(n) = (prime(2*m)-prime(m))/6 where m is the n-th positive integer such that (prime(2*m)-prime(m))/6 is prime.
%H A141657 David A. Corneth, <a href="/A141657/b141657.txt">Table of n, a(n) for n = 1..10000</a>
%e A141657 For m=4,  (p(2*4)  - p(4))/6  =  (19 -  7)/6 =  2 = a(1).
%e A141657 For m=5,  (p(2*5)  - p(5))/6  =  (29 - 11)/6 =  3 = a(2).
%e A141657 For m=10, (p(2*10) - p(10))/6 =  (71 - 29)/6 =  7 = a(3).
%e A141657 For m=15, (p(2*15) - p(15))/6 = (113 - 47)/6 = 11 = a(4).
%e A141657 For m=16, (p(2*16) - p(16))/6 = (131 - 53)/6 = 13 = a(5), etc.
%t A141657 Select[Table[(Prime[2n]-Prime[n])/6,{n,1000}],PrimeQ[#]&] (* _Harvey P. Dale_, Jun 18 2023 *)
%Y A141657 Cf. A000040, A072473.
%K A141657 nonn
%O A141657 1,1
%A A141657 _Juri-Stepan Gerasimov_, Sep 18 2008
%E A141657 19, 23 inserted, one instance of 331 removed, 577 removed and extended by _R. J. Mathar_, Oct 04 2008
%E A141657 Definition clarified by _Harvey P. Dale_, Jun 18 2023
%E A141657 Definition clarified by _David A. Corneth_, Jun 18 2023