A306487 - OEIS

%I #25 Mar 13 2019 23:49:06

%S 561,645,1105,1729,1905,2465,2821,4371,6601,8481,8911,10585,11305,

%T 12801,13741,13981,15841,16705,18705,23001,25761,29341,30121,30889,

%U 33153,34945,39865,41041,41665,46657,52633,55245,57421,62745,63973,68101,72885,74665,75361

%N Poulet numbers which are not super-Poulet numbers.

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

%C Is this sequence infinite?

%C 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.

%D 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.

%H Antti Karttunen, <a href="/A306487/b306487.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PouletNumber.html">Poulet Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Super-PouletNumber.html">Super-Poulet Numbers</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Super-Poulet_number">Super-Poulet number</a>

%e 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

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

%o (PARI)

%o is_A001567(n) = {Mod(2, n)^(n-1)==1 && !isprime(n) && n>1}; \\ From A001567 by _M. F. Hasler_

%o 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.

%o is_A306487(n) = (is_A001567(n) && !is_A050217(n)); \\ (Probably could be reduced to a simpler program). - _Antti Karttunen_, Feb 28 2019

%o (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

%Y Cf. A001567, A050217, A006935, A080747.

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

%K nonn

%O 1,1

%A _Amiram Eldar_, Feb 18 2019