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A143584
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Integers that are equal to the multiplicative order of 2 modulo some overpseudoprime to base 2.
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2
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11, 23, 25, 28, 29, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 63, 64, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 79, 81, 82, 83, 84, 87, 88, 91, 92, 94, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110, 111, 112
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OFFSET
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1,1
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COMMENTS
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A064078(a(n)) is a composite number. The sequence has a positive density since it contains, in particular, numbers of the form 8n+20 for n >= 1 (C. Pomerance, private correspondence). Since, e.g., 38 is not in the sequence, there is not an overpseudoprime m such that ord_m(2)=38.
Phi_{a(n)}(2), the a(n)-th cyclotomic polynomial of x evaluated at x=2 has at least 2 distinct prime factors that are not prime factors of the Phi_k(2) for any positive integer k < a(n). For example, Phi_11(2) = 2^11 - 1 = 2047 = 23 * 89 and Phi_25(2) = 2^20 + 2^15 + 2^10 + 2^5 + 1 = 1082401 = 601 * 1801. Note that p = a(n) is prime if and only if Phi_p(2) = 2^p - 1 is composite. - David Terr, Sep 09 2018
It is easy to prove the statement above. We use the fact that Phi_j(n) and Phi_k(n) are coprime whenever j and k are coprime as well as the fact that an overpseudoprime has at least 2 distinct prime factors. - David Terr, Oct 10 2018
A number k is included iff either 2^k-1 has more than one primitive prime factor (cf. A086251, A161508) or the only primitive prime factor of 2^k-1 is a Wieferich prime (no examples known). - Jeppe Stig Nielsen, Sep 01 2020
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LINKS
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PROG
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(PARI) isok(k) = my(m=polcyclo(k, 2)); m/=gcd(m, k); m!=1&&!isprime(m) \\ Jeppe Stig Nielsen, Sep 01 2020
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CROSSREFS
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Cf. A131952 (for the corresponding maximal overpseudoprimes).
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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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