A175737 -id:A175737 - cp's OEIS frontend

A175736 Increasing gaps between 2-pseudoprimes (lower end).

341, 645, 3277, 4681, 5461, 6601, 11305, 16705, 19951, 25761, 35333, 68101, 75361, 93961, 137149, 294409, 367081, 688213, 950797, 1207361, 2035153, 2537641, 3235699, 6386993, 7040001, 12096613, 19781763, 24356377, 34944001, 49664207, 50523661

Offset: 1

Charles R Greathouse IV, Aug 28 2010

nonn

A175738 is the main entry for this sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..207
Index entries for sequences related to pseudoprimes

Cf. A175737 (upper end), A175738.

A175738 Sizes of successive increasing gaps between 2-pseudoprimes.

Original entry on oeis.org

220, 460, 756, 780, 1140, 1356, 1496, 2000, 3050, 3580, 4532, 4784, 5220, 7140, 12132, 20412, 20650, 22320, 26076, 39424, 42392, 65740, 81252, 87698, 137104, 164448, 203506, 370396, 484140, 491526, 506940, 667908, 682820, 777224, 951114, 1201538

Offset: 1

Charles R Greathouse IV, Aug 28 2010

Rotkiewicz proves that a(n) < A175736(n)^2, and that the exponent can be replaced by 1 + epsilon for large enough n.

A. Rotkiewicz, "Les intervalles contenants les nombres pseudopremiers", Rendiconti del Circolo Matematico di Palermo, Series 2, 14 (1965), pp. 278-280.

Charles R Greathouse IV, Table of n, a(n) for n = 1..207
Charles R Greathouse IV, Illustration of gap sizes vs. lower end
H. Halberstam and A. Rotkiewicz, A gap theorem for pseudoprimes in arithmetic progression, Acta Arithmetica 13 (1967/1968), pp. 395-404.
Eric Weisstein's World of Mathematics, Fermat Pseudoprime
Index entries for sequences related to pseudoprimes

Cf. A001567 (2-pseudoprimes), A175736 (lower end), A175737 (upper end).

a(n) = A175737(n) - A175736(n).

A335326 Lesser of 2 consecutive Fermat pseudoprimes to base 2 with no prime numbers in between them.

Original entry on oeis.org

4369, 13741, 31609, 6973057, 208969201

Offset: 1

Amiram Eldar, Jun 01 2020

The corresponding larger counterparts are 4371, 13747, 31621, 6973063, 208969223.
The corresponding gaps between the consecutive pseudoprimes are 2, 6, 12, 6, 22.
a(6) > 10^15, if it exists.

			4369 is a term since it is a Fermat pseudoprime to base 2, and between it and the next Fermat pseudoprime to base 2, 4371, there is only one number, 4370, which is composite.

Cf. A001567, A173619, A175736, A175737, A175738.

psp = Cases[Import["https://oeis.org/A001567/b001567.txt", "Table"], {, }][[;; , 2]]; s = {}; Do[If[NextPrime[psp[[k]]] > psp[[k + 1]], AppendTo[s, psp[[k]]]], {k, 1, Length[psp] - 1}]; s

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A175736 Increasing gaps between 2-pseudoprimes (lower end).

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A175738 Sizes of successive increasing gaps between 2-pseudoprimes.

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A335326 Lesser of 2 consecutive Fermat pseudoprimes to base 2 with no prime numbers in between them.

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