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.

单词	SuperPouletNumber
释义	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.
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