A308987 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.

1, 2, 4, 10, 14, 16, 20, 34, 40, 46, 88, 100, 112, 130, 152, 212, 288, 330, 346, 444, 502, 526, 534, 564, 580, 614, 624, 634, 636, 640, 690

Offset: 1

Author

Trevor Cappallo, Jul 04 2019

Keywords

• nonn
• more

Comments

This sequence represents the highest gaps, given by number of terms (including the starting prime) in sequence A002522 between terms which are prime.

Examples

			n=6   -->  6^2+1 = 37, prime
n=7   -->  7^2+1 = 50, composite
n=8   -->  8^2+1 = 65, composite
n=9   -->  9^2+1 = 82, composite
n=10  -->  10^2+1 = 101, prime
...so here m=3 and we get the third term, m + 1 = 10 - 6 = 4
		

Crossrefs

Cf. A002496, A002522, A308988.

A293564 gives essentially the same information.

Programs

Mathematica

best = c = lastBestAt = 0;
For[i = 2, True, i += 2; c += 2,
If[PrimeQ[i^2 + 1],
   If[c > best,
    best = c;
    bestAt = i - c;
    If[bestAt != lastBestAt, Print[{c, bestAt}]];
    lastBestAt = bestAt;
    ];
   c = 0;
   ]
]
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 *)

Extensions

a(21)-a(31) from Giovanni Resta, Jul 05 2019