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A216364 Fermat pseudoprimes to base 2 divisible by 15.

%I A216364 #22 Feb 16 2025 08:33:18
%S A216364 645,1905,18705,55245,62745,72885,215265,451905,831405,1246785,
%T A216364 1472505,1489665,1608465,1815465,2077545,2113665,2882265,4535805,
%U A216364 6135585,6242685,8322945,9063105,9816465,16263105,18137505,19523505,53661945,63560685,81612105,81722145
%N A216364 Fermat pseudoprimes to base 2 divisible by 15.
%C A216364 Most of the numbers in the sequence above can be written in one of just two forms: 15*(42*n + 1) and 15*(42*n - 13):
%C A216364 (I) numbers of the first form and the corresponding n in the brackets: 645(1), 1905(3), 1246785(1979), 2113665(3355), 2882265(4575), 6135585(9739); 6242685(9909); 8322945(13211), 81612105(129543);
%C A216364 (II) numbers of the second form and the corresponding n in the brackets: 18705(30), 55245(88), 72885(116), 215265(342), 831405(1320), 1815465(2882), 2077545(3298), 4535805(7200), 9816465(15582), 18137505(28790), 19523505(30990), 53661945(85178), 81722145(129718).
%C A216364 But these pseudoprimes can be categorized in many ways taking, beside 42, p - 1, where p is a prime divisor common to many of them (e.g., numbers of the form 15*(46*n + 43) and the corresponding n in the brackets: 62745 (90); 451905 (654); 1489665(2158); 9063105(13134); 63560685(92116)) or p + 1 (e.g., numbers of the form 15*(90*n + 67) and the corresponding n in the brackets: 1472505(1090); 16263105(12046)).
%C A216364 What is also interesting about these numbers: the Fermat pseudoprimes to base 2 formed with their prime divisors, different from 3 and 5 (e.g., 645 = 15*43 and 1905 = 15*127) are Fermat pseudoprimes to base 2, but also 5461 = 43*127; 18705 = 15*29*43 and 55245 = 15*29*127 are Fermat pseudoprimes to base 2, and 158369 = 29*43*127.
%C A216364 Note: Fermat pseudoprimes to base 2 divisible by 5 are mostly of the form 3*k or 3*k + 1; of the first 100 numbers divisible by 5 checked, fewer than 10 are of the form 3*k + 2.
%H A216364 Amiram Eldar, <a href="/A216364/b216364.txt">Table of n, a(n) for n = 1..10000</a>
%H A216364 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>
%t A216364 Select[15*Range[10^6], PowerMod[2, # - 1, #] == 1 &] (* _Amiram Eldar_, Mar 07 2020 *)
%o A216364 (PARI) is_a216364(n) = {Mod(2, n)^n==2 & !isprime(n) & Mod(n, 15)==0} \\ _Michael B. Porter_, Jan 27 2013
%Y A216364 Intersection of A001567 and A008597.
%Y A216364 Cf. A215672, A215944, A216023.
%K A216364 nonn
%O A216364 1,1
%A A216364 _Marius Coman_, Sep 05 2012