A218010 Primes of the form (24*p + 1)/5, where p is a Fermat pseudoprime to base 2.

1637, 2693, 20981, 22469, 40709, 42773, 49253, 65957, 69557, 123653, 140837, 235877, 451013, 623621, 626693, 716549, 1095557, 1370597, 1634693, 1761989, 2289461, 2459813, 2548229, 2563493, 2821733, 3414533, 4091909, 4093637, 4910981, 5530901, 5727461

Offset

1,1

Comments

This is a subsequence of A107003.

The corresponding values of p: 341, 561, 4371, 4681, 8481, 8911, 10261, 13741, 14491, 25761, 29341, 49141, 93961, 129921, 130561, 149281, 228241, 285541, 340561, 439291, 512461, 530881, 532171, 534061, 597871, 736291, 764491, 782341, 852841, 903631, 951481.

From the first 128 natural solutions of this equation ((24*p + 1)/5, where p is Fermat pseudoprime to base 2), 31 are primes (the ones from the sequence above), 51 are products (not necessarily squarefree) of two prime factors and 41 are products of three prime factors; only 5 of them are products of four prime factors.

Conjecture: There is no absolute Fermat pseudoprime m for which n = (5*m - 1)/24 is a natural number (checked for the first 300 Carmichael numbers; if true, then the formula is a criterion to separate pseudoprimes at least from a subset of primes, because there are 37 primes m from the first 300 primes for which n = (5*m - 1)/24 is a natural number).

3380740301 is a counterexample to the conjecture. - Charles R Greathouse IV, Dec 07 2014

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Fermat Pseudoprime

Prog

(PARI) is(n)=my(t); n%48==5 && isprime(n) && !isprime(t=(5*n-1)/24) && Mod(2, t)^t==2 \\ Charles R Greathouse IV, Dec 07 2014

Crossrefs

Cf. A107003, A142399.

Sequence in context: A224461 A233121 A054808 * A298075 A362405 A239160

Adjacent sequences: A218007 A218008 A218009 * A218011 A218012 A218013

Keyword

nonn

Author

Marius Coman, Oct 18 2012

Extensions

Corrected by Charles R Greathouse IV, Dec 07 2014

Status

approved