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A144755
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Primes which divide none of overpseudoprimes to base 2 (A141232).
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9
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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Overpseudoprimes to base 2 are odd, then a(1)=2.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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