cp's OEIS Frontend

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.

Old name was: Fermat pseudoprimes to base 2 of the form 8*p*n + p^2, where p is prime and n natural.

For p = 5 the formula becomes 40*n + 25. From the first 15 pseudoprimes divisible by 5, 12 are of the form 40*n + 25 (beside 3 of them which are of the form 40*n + 5). Conjecture: there are no pseudoprimes to base 2 of the form 40*n + 15.

Note: it can be seen that a pseudoprime can be written in this formula in more than one way: e.g., 561 = 8*3*23 + 3^2 = 8*11*5 + 11^2 = 8*17*2 + 17^2 or 1905 = 8*3*79 + 3^2 = 8*5*47 + 5^2.

Conjecture: If a Fermat pseudoprime to base 2 can be written as 8*p*n + p^2, where n is an integer and p one of its prime factors, then it can be written this way for any of its prime factors. Checked for all pseudoprimes from the sequence above.

Conjecture: If a Fermat pseudoprime to base 2 with two prime factors can be written as 8*p1*n + p1^2, where n is a natural number and p1 one of its two prime factors, then it can also be written as 8*p2*(-n) + p2^2, where p2 is the other prime factor. Checked for 4033 = 37*109(n = 9), 4369 = 17*257(n = 30), 4681 = 31*151(n = 15), 8321 = 53*157(n = 13), 18721 = 97*193(n = 12), 23377 = 97*241(n = 18), 31417 = 89*353(n = 33), 31609 = 73*433 (n = 45), 65281 = 97*673(n = 72), 85489 = 53*1613 (n = 195).

Conjecture: If a Fermat pseudoprime to base 2 cannot be written as 8*p*n + p^2, where n is an integer and p one of its prime factors, then it cannot be written this way for any of its prime factors. Checked for the following pseudoprimes: 341, 645, 1387, 2047, 2701, 2821, 3277, 4371, 5461, 7957, 10261, 13741, 13747, 13981, 14491, 15709, 19951, 29341, 31621, 42799, 49141, 49981, 55245, 60701, 60787, 63973, 65077, 68101, 72885, 80581, 83333.

Note: from the first 72 pseudoprimes, 39 can be written this way.

All three conjectures are true (obvious from new characterization). - Charles R Greathouse IV, Dec 07 2014