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A306487 Poulet numbers which are not super-Poulet numbers. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)