super-Poulet number

A super-Poulet number $n$ is a Poulet number which besides satisfying the congruence $2^{n}\equiv 2\mod n$ , each of its divisors $d_{i}$ (for $1<i\leq\tau(n)$ ) also satisfies the congruence $2^{d_{i}}\equiv 2\mod d_{i}$ .

Two examples: 341 is a super-Poulet number, with its divisors being 1, 11, 31 and 341 itself. We verify that $2^{11}=2048=11\times 186+2$ and $2^{31}=2147483648=31\times 69273666+2$ . 341 itself has already been checked when confirmed as a Poulet number. Now, 561 is a Poulet number but not a super-Poulet number since one of its divisors, 33, does not satisfy the congruence: $\frac{2^{33}-2}{33}\approx 260301048.18181818\ldots$ .

The first few super-Poulet numbers are 341, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, which are listed in A050217 of Sloane’s OEIS.

Title	super-Poulet number
Canonical name	SuperPouletNumber
Date of creation	2013-03-22 18:14:12
Last modified on	2013-03-22 18:14:12
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A51