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%I #13 Mar 05 2022 14:18:36
%S 13,97,113,193,283,397,479,673,953,1439,1597,2297,2699,3469,4219,4483,
%T 5323,7219,8273,9209,9403,10799,12097,13219,14879,15373,15619,21313,
%U 23399,26237,27883,32029,32749,34217,37243,39989,41203,42433,43669,46219,55219,60509,62497
%N Lesser of consecutive primes whose average is of the form k*(k+2), for some integer k.
%e 193 is in the sequence because 193 is prime, nextprime(193) = 197, and (193+197)/2 = 195 = 13*(13+2).
%t Select[Partition[Prime[Range[6500]],2,1],IntegerQ[SolveValues[(k(k+2)) == Mean[#],k][[2]]]&][[All,1]] (* _Harvey P. Dale_, Mar 05 2022 *)
%o (PARI) {k=2;while(k<10^5,l=nextprime(k+1);if(issquare((k+l)/2+1),print1(k,", "));k=l)}
%Y Cf. A242384.
%K nonn
%O 1,1
%A _Antonio Roldán_, May 12 2014
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