A306487 Poulet numbers which are not super-Poulet numbers.

561, 645, 1105, 1729, 1905, 2465, 2821, 4371, 6601, 8481, 8911, 10585, 11305, 12801, 13741, 13981, 15841, 16705, 18705, 23001, 25761, 29341, 30121, 30889, 33153, 34945, 39865, 41041, 41665, 46657, 52633, 55245, 57421, 62745, 63973, 68101, 72885, 74665, 75361

Offset

1,1

Comments

Subsequence of A080747 from which this differs for the first time at n=78, with A080747(78) = 294409, a term not present here.

Is this sequence infinite?

According to Sierpinski there are infinitely many Poulet numbers which are not super-Poulet numbers. But his definition of Poulet numbers includes the even pseudoprimes to base 2 (A006935), and the proof is based on the infinitude of this sequence and that super-Poulet numbers are never even.

References

W. Sierpinski, Elementary Theory of Numbers, ed. A. Schinzel, North-Holland Mathematical Library (2nd ed.), Amsterdam: North Holland, 1988, Chapter V, p. 234, Exercise 1.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Poulet Number

Eric Weisstein's World of Mathematics, Super-Poulet Numbers

Wikipedia, Super-Poulet number

Example

561 is in the sequence because 2^561 % 561 == 2 but 33|561 and 2^33 % 33 = 8 <> 2. - David A. Corneth, Feb 28 2019

Mathematica

Select[Select[Range[3, 100000, 2], !PrimeQ[ # ] && PowerMod[2, (# - 1), # ] == 1 &], Union[PowerMod[2, Rest[Divisors[#]], #]] != {2}& ]

Prog

(PARI)
is_A001567(n) = {Mod(2, n)^(n-1)==1 && !isprime(n) && n>1}; \\ From A001567 by M. F. Hasler
is_A050217(n) = if(isprime(n), 0, fordiv(n, d, if(Mod(2, d)^d!=2, return(0))); (n>1)); \\ After Charles R Greathouse IV's Aug 27 2016 PARI-program in A050217.
is_A306487(n) = (is_A001567(n) && !is_A050217(n)); \\ (Probably could be reduced to a simpler program). - Antti Karttunen, Feb 28 2019
(PARI) is(n) = {if(isprime(n) || n < 2 || n%2 == 0, return(0)); if(Mod(2, n)^n!=2, return(0) , d = divisors(n); for(i = 1, #d-1, if(Mod(2, d[i])^d[i]!=2, return(1) ) ) ); 0 } \\ David A. Corneth, Feb 28 2019

Crossrefs

Cf. A001567, A050217, A006935, A080747.

Cf. A215672 (differs from a(13) = 11305 on, which is not in A215672).

Sequence in context: A259172 A192297 A080747 * A074380 A215672 A085999

Adjacent sequences: A306484 A306485 A306486 * A306488 A306489 A306490

Keyword

nonn

Author

Amiram Eldar, Feb 18 2019

Status

approved