A218010 - OEIS

%I #12 Jan 29 2023 20:26:22

%S 1637,2693,20981,22469,40709,42773,49253,65957,69557,123653,140837,

%T 235877,451013,623621,626693,716549,1095557,1370597,1634693,1761989,

%U 2289461,2459813,2548229,2563493,2821733,3414533,4091909,4093637,4910981,5530901,5727461

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

%C This is a subsequence of A107003.

%C 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.

%C 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.

%C 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).

%C 3380740301 is a counterexample to the conjecture. - _Charles R Greathouse IV_, Dec 07 2014

%H Charles R Greathouse IV, <a href="/A218010/b218010.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatPseudoprime.html">Fermat Pseudoprime</a>

%o (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

%Y Cf. A107003, A142399.

%K nonn

%O 1,1

%A _Marius Coman_, Oct 18 2012

%E Corrected by _Charles R Greathouse IV_, Dec 07 2014