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A216364 Fermat pseudoprimes to base 2 divisible by 15. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

Table of n, a(n) for n=1..30.

Eric Weisstein's World of Mathematics, Poulet Number

PROG

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

CROSSREFS

Cf. A001567, A215672, A215944, A216023.

Sequence in context: A216023 A100873 A227136 * A063844 A265684 A067845

Adjacent sequences:  A216361 A216362 A216363 * A216365 A216366 A216367

KEYWORD

nonn

AUTHOR

Marius Coman, Sep 05 2012

STATUS

approved

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Last modified May 19 02:29 EDT 2019. Contains 323377 sequences. (Running on oeis4.)