A215343 Fermat pseudoprimes to base 2 that can be written as 2*p^2 - p, where p is also a Fermat pseudoprime to base 2.

831405, 5977153, 15913261, 21474181, 38171953, 126619741, 210565981, 224073865, 327718401, 377616421, 390922741, 558097345, 699735345, 700932961, 1327232481, 1999743661, 4996150741, 8523152641, 11358485281, 13999580785, 15613830541, 17657245081, 20442723301

Offset: 1

Author

Marius Coman, Aug 08 2012

Keywords

• nonn

Comments

Fermat pseudoprimes are listed in A001567.

The correspondent p for the numbers from the sequence above: 645, 1729, 2821, 3277, 4369, 7957, 10261, 10585, 12801, 13741, 13981, 16705, 18705, 25761, 31621, 49981, 65281, 75361, 83665, 88357, 93961, 101101.

Note that for 22 of the first 80 Poulet numbers, we obtained through this formula another Poulet number!

The formula could be generalized this way: Poulet numbers that can be written as (n + 1)*p^2 - n*p, where n is natural, n > 0, and p is another Poulet number.

For n = 1, that formula becomes the formula set out for the sequence above.

For n = 2, that formula becomes 3*p^2 - 2*p, from which the Poulet numbers 348161 (for p = 341) and 1246785 (for p = 645) were obtained.

For n = 3, that formula becomes 4*p^2 - 3*p, from which the Poulet number 119273701 (for p = 5461) was obtained.

For n = 4, that formula becomes 5*p^2 - 4*p, from which the Poulet numbers 2077545 (for p = 645) and 9613297 (for p = 1387) were obtained.

Conjecture: there are infinitely many Poulet numbers that can be written as (n + 1)*p^2 - n*p, where n is natural, n > 0, and p is another Poulet number.

Finally, considering, e.g., that for the Poulet number 645, Poulet numbers were obtained for n = 1, 2, 4 (i.e., 831405, 1246785, 2077545), yet another conjecture: For any Poulet number p, there are infinitely many Poulet numbers that can be written as (n + 1)*p^2 - n*p, where n is natural, n > 0.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Poulet Number

Crossrefs

Cf. A001567, A213812.

Extensions

Edited by Jon E. Schoenfield, Dec 12 2013

a(14) inserted by Charles R Greathouse IV, Jul 07 2017