Poulet numbers

According to Fermat's little theorem if $p$

is an odd prime number then $2^{p-1}-1$ (or, equivalently, $2^p-2$

) is divisible by $p$

A number 1$"> which is not prime, but such that $2^n-2$ is divisible by $n$ is called a base-2 Fermat pseudoprime, or simply a Poulet or Sarrus number.

Up to $10^{15}$ there are 1801533 Poulet numbers.

The first Poulet numbers are 341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many Poulet numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Poulet Number
Wikipedia, Poulet number
OEIS, Sequence A001567
W. Galway, Tables of pseudoprimes and related data (Pseudoprimes up to 2⁶⁴).

Poulet numbers can also be... (you may click on names or numbers and on + to get more values)

aban 341 561 645 + 949275000833 alternating 341 561 2701 + 967270129 amenable 341 561 645 + 998724481 apocalyptic 1105 2047 2701 + 29341 arithmetic 341 561 645 + 9995671 binomial 561 2701 4371 + 9999899578441 brilliant 341 1387 2047 + 998590601 c.decagonal 2704801 4154161 5859031 + 248626476315361 c.heptagonal 5461 19951 49981 + 180154037487061 c.nonagonal 2701 8911 10585 + 999961448323753 c.octagonal 1194649 12327121 c.pentagonal 219781 399001 2537641 + 18823846860001 c.square 1105 8321 107185 + 444733131907121 c.triangular 30889 512461 9371251 + 597565005525001 Carmichael 561 1105 1729 + 999629786233 congruent 341 561 645 + 9995671 constructible 4369 16843009 Cunningham 1729 2047 46657 + 598865079758401 Curzon 341 561 645 + 181285001 cyclic 341 561 1105 + 9920401 d-powerful 2465 194221 223345 + 9567673 de Polignac 2465 31417 41665 + 97255801 decagonal 1105 1387 2047 + 999989162797327 deceptive 1729 2821 5461 + 99976607641 deficient 341 561 645 + 9995671 dig.balanced 2701 7957 8321 + 197747377 Duffinian 341 1105 1387 + 9995671 economical 1387 2047 2701 + 19985269 emirpimes 341 1387 2047 + 99789673 equidigital 1387 2047 2701 + 19985269 eRAP 34945 evil 561 645 1105 + 999828727 Friedman 41665 46657 frugal 1194649 12327121 gapful 341 561 1729 + 99909442585 happy 12801 18705 60701 + 9439201 Harshad 645 1387 1729 + 9640119601 heptagonal 23377 164737 401401 + 999953670536617 hex 1387 4681 7957 + 916540502502241 hexagonal 561 2701 4371 + 999961448323753 hoax 645 4369 13747 + 97863529 Hogben 4033 25761 65281 + 421925923038061 iban 341 2047 2701 + 722201 inconsummate 645 4371 8481 + 769757 interprime 645 7957 11305 + 91453705 Jacobsthal 341 5461 1398101 + 46912496118443 junction 8321 11305 126217 + 99830641 Kaprekar 670033 katadrome 8321 Lehmer 561 1105 1387 + 999986341201 lucky 645 1105 1387 + 9920401 magic 1105 2465 6998881 139101047324161 magnanimous 2465 6601 modest 35333 83333 256999 + 799808401 Moran 645 1387 1905 nialpdrome 8321 83333 653333 + 87777776521 nonagonal 129889 2288661 4650049 + 999971076292381 octagonal 341 645 2465 + 999993874626581 odious 341 1387 1729 + 998724481 palindromic 101101 129921 1837381 pancake 5461 1493857 5250421 + 56296655194501 pandigital 39865 212421 5258701 + 8137633 pentagonal 7957 241001 1419607 + 999894027031901 pernicious 341 1387 1729 + 9995671 persistent 15076432489 17392546081 25089467413 + 92345876701 plaindrome 23377 223345 256999 1333333 power 1194649 12327121 powerful 1194649 12327121 Proth 1729 4033 8321 + 137439477761 repunit 341 2047 4033 + 421925923038061 Ruth-Aaron 182356993 2320690177 3203895601 + 867347602001 self 4371 15709 31417 + 997753901 self-describing 33193117 semiprime 341 1387 2047 + 99945007 Smith 645 4369 13747 + 97863529 sphenic 561 645 1105 + 99971821 square 1194649 12327121 star 49141 976873 2649361 + 175321803251881 strobogrammatic 101101 super-d 4369 4681 8481 + 9890881 taxicab 1729 23226658794001 105193759783861 tetrahedral 6891657409 777080801185 1581289265305 + 674228077807189 triangular 561 2701 4371 + 999961448323753 uban 17002089000001 Ulam 341 1387 14491 + 9774181 unprimeable 11305 74665 223345 + 9816465 wasteful 341 561 645 + 9995671 Woodall 2047 137438953471 Zeisel 1729 294409 2953711 + 959377262271049