Trevor Cappallo has authored 2 sequences.
Original entry on oeis.org
1, 2, 6, 26, 40, 94, 184, 350, 496, 3390, 3536, 45370, 82734, 99064, 357164, 840904, 3880556, 27914936, 40517520, 104715206, 1126506904, 2084910530, 2442825346, 4332318176, 6716598046, 17736392220, 18205380336, 30869303806, 68506021364, 78491213264, 85620067844
Offset: 1
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
...yields the third record gap number of terms, so the start index n=6 appears as the third entry in this sequence.
-
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;
]
]
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.
Original entry on oeis.org
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
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
A293564 gives essentially the same information.
-
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 *)
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