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Homesuper-Poulet number

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# 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.

## Mathematics Subject Classification

11A51*no label found*

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