A144755 Primes which divide none of overpseudoprimes to base 2 (A141232).

2, 3, 5, 7, 11, 13, 17, 19, 31, 41, 43, 73, 127, 151, 241, 257, 331, 337, 683, 2731, 5419, 8191, 43691, 61681, 65537, 87211, 131071, 174763, 262657, 524287, 599479, 2796203, 15790321, 18837001, 22366891, 715827883, 2147483647, 4278255361

Offset: 1

Author

Vladimir Shevelev, Sep 20 2008

Keywords

• nonn

Comments

Odd prime p is in the sequence iff A064078(A002326((p-1)/2))=p. For example, for p=127 we have A002326((127-1)/2)=7 and A064078(7)=127. Thus p=127 is in the sequence.

Primes p such that the binary expansion of 1/p has a unique period length; that is, no other prime has the same period. Sequence A161509 sorted. - T. D. Noe, Apr 13 2010

Since A161509 has terms of varying magnitude, sorting any finite initial segment of A161509 cannot provide a guarantee that there are no other terms missed in between. Any prime p not (yet) appearing in A161509 should be tested via A064078(A002326((p-1)/2))=p to conclude whether it belongs to the current sequence. - Max Alekseyev, Feb 10 2024

Examples

			Overpseudoprimes to base 2 are odd, then a(1)=2.
		

Crossrefs

Cf. A141232, A002326, A064078, A112927.

Cf. A040017 (unique-period primes in base 10). - T. D. Noe, Apr 13 2010

Programs

Mathematica

b=2; t={}; Do[c=Cyclotomic[n,b]; q=c/GCD[n,c]; If[PrimePowerQ[q], p=FactorInteger[q][[1,1]]; If[p<10^12, AppendTo[t,p]; Print[{n,p}]]], {n,1000}]; t=Sort[t] (* T. D. Noe, Apr 13 2010 *)

PARI

{ is_a144755(p) = my(q,m,g); q=znorder(Mod(2,p)); m=2^q-1; fordiv(q,d, if(d1,m\=g))); m==p; } \\ Max Alekseyev, Feb 10 2024

Extensions

Extended by T. D. Noe, Apr 13 2010

b-file deleted by Max Alekseyev, Feb 10 2024.