A216364 Fermat pseudoprimes to base 2 divisible by 15.

645, 1905, 18705, 55245, 62745, 72885, 215265, 451905, 831405, 1246785, 1472505, 1489665, 1608465, 1815465, 2077545, 2113665, 2882265, 4535805, 6135585, 6242685, 8322945, 9063105, 9816465, 16263105, 18137505, 19523505, 53661945, 63560685, 81612105, 81722145

Offset: 1

Author

Marius Coman, Sep 05 2012

Keywords

• nonn

Comments

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):

(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);

(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).

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)).

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.

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Poulet Number

Crossrefs

Intersection of A001567 and A008597.

Cf. A215672, A215944, A216023.

Programs

Mathematica

Select[15*Range[10^6], PowerMod[2, # - 1, #] == 1 &] (* Amiram Eldar, Mar 07 2020 *)

PARI

is_a216364(n) = {Mod(2, n)^n==2 & !isprime(n) & Mod(n, 15)==0} \\ Michael B. Porter, Jan 27 2013