A308987 - OEIS

%I #19 Sep 27 2024 17:25:59

%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 ]

%t Join[{1,2},Rest[DeleteDuplicates[Length/@SplitBy[(Range[5*10^7]^2+1),PrimeQ],GreaterEqual]+1]] (* The program generates the first 19 terms of the sequence. *)(* _Harvey P. Dale_, Sep 27 2024 *)

%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