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

%I A144755 #17 Feb 11 2024 01:05:15
%S A144755 2,3,5,7,11,13,17,19,31,41,43,73,127,151,241,257,331,337,683,2731,
%T A144755 5419,8191,43691,61681,65537,87211,131071,174763,262657,524287,599479,
%U A144755 2796203,15790321,18837001,22366891,715827883,2147483647,4278255361
%N A144755 Primes which divide none of overpseudoprimes to base 2 (A141232).
%C A144755 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.
%C A144755 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
%C A144755 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
%e A144755 Overpseudoprimes to base 2 are odd, then a(1)=2.
%t A144755 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 *)
%o A144755 (PARI) { is_a144755(p) = my(q,m,g); q=znorder(Mod(2,p)); m=2^q-1; fordiv(q,d, if(d<q, while((g=gcd(m,2^d-1))>1,m\=g))); m==p; } \\ _Max Alekseyev_, Feb 10 2024
%Y A144755 Cf. A141232, A002326, A064078, A112927.
%Y A144755 Cf. A040017 (unique-period primes in base 10). - _T. D. Noe_, Apr 13 2010
%K A144755 nonn
%O A144755 1,1
%A A144755 _Vladimir Shevelev_, Sep 20 2008
%E A144755 Extended by _T. D. Noe_, Apr 13 2010
%E A144755 b-file deleted by _Max Alekseyev_, Feb 10 2024.