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%I #17 Aug 03 2019 22:03:37
%S 1,2,4,10,14,16,20,34,40,46,88,100,112,130,152,212,288,330,346,444,
%T 502,526,534,564,580,614,624,634,636,640,690
%N In the sequence {n^2+1} (A002522), color the primes red. When the number of terms m between successive red terms sets a new record, write down m+1.
%C This sequence represents the highest gaps, given by number of terms (including the starting prime) in sequence A002522 between terms which are prime.
%e n=6 --> 6^2+1 = 37, prime
%e n=7 --> 7^2+1 = 50, composite
%e n=8 --> 8^2+1 = 65, composite
%e n=9 --> 9^2+1 = 82, composite
%e n=10 --> 10^2+1 = 101, prime
%e ...so here m=3 and we get the third term, m + 1 = 10 - 6 = 4
%t best = c = lastBestAt = 0;
%t For[i = 2, True, i += 2; c += 2,
%t If[PrimeQ[i^2 + 1],
%t If[c > best,
%t best = c;
%t bestAt = i - c;
%t If[bestAt != lastBestAt, Print[{c, bestAt}]];
%t lastBestAt = bestAt;
%t ];
%t c = 0;
%t ]
%t ]
%Y Cf. A002496, A002522, A308988.
%Y A293564 gives essentially the same information.
%K nonn,more
%O 1,2
%A _Trevor Cappallo_, Jul 04 2019
%E a(21)-a(31) from _Giovanni Resta_, Jul 05 2019
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