cp's OEIS Frontend

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

%I A218010 #13 Feb 16 2025 08:33:18
%S A218010 1637,2693,20981,22469,40709,42773,49253,65957,69557,123653,140837,
%T A218010 235877,451013,623621,626693,716549,1095557,1370597,1634693,1761989,
%U A218010 2289461,2459813,2548229,2563493,2821733,3414533,4091909,4093637,4910981,5530901,5727461
%N A218010 Primes of the form (24*p + 1)/5, where p is a Fermat pseudoprime to base 2.
%C A218010 This is a subsequence of A107003.
%C A218010 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 A218010 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 A218010 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 A218010 3380740301 is a counterexample to the conjecture. - _Charles R Greathouse IV_, Dec 07 2014
%H A218010 Charles R Greathouse IV, <a href="/A218010/b218010.txt">Table of n, a(n) for n = 1..10000</a>
%H A218010 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatPseudoprime.html">Fermat Pseudoprime</a>
%o A218010 (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 A218010 Cf. A107003, A142399.
%K A218010 nonn
%O A218010 1,1
%A A218010 _Marius Coman_, Oct 18 2012
%E A218010 Corrected by _Charles R Greathouse IV_, Dec 07 2014