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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
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
1,2
COMMENTS
This sequence represents the highest gaps, given by number of terms (including the starting prime) in sequence A002522 between terms which are prime.
EXAMPLE
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
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 *)
CROSSREFS
A293564 gives essentially the same information.
Sequence in context: A034233 A056718 A057283 * A107992 A139480 A227388
KEYWORD
nonn,more
AUTHOR
Trevor Cappallo, Jul 04 2019
EXTENSIONS
a(21)-a(31) from Giovanni Resta, Jul 05 2019
STATUS
approved