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#26 by Jianing Song at Sat Dec 24 05:53:04 EST 2022
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An odd prime p is a term if and only if p == 3 (mod 4), and that the multiplicative order of 2 modulo p is p-1 or (p-1)/2 (p-1 is p == 3 (mod 8), (p-1)/2 if p == 7 (mod 8)).
{(a(n)-1)/2} is the sequence of indices of fixed points of A302141.
An odd prime p is a term if and only if p == 3 (mod 4), and that the multiplicative order of 2 modulo p is p-1 or (p-1)/2 (p-1 is p == 3 (mod 8), (p-1)/2 if p == 7 (mod 8)).
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proposed
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#25 by Jianing Song at Sat Dec 24 05:49:04 EST 2022
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#24 by Jianing Song at Sat Dec 24 05:48:30 EST 2022
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Also primes p such that the multiplicative order of 16 modulo p is (p-1)/2. Proof: note that ord(16,p) = ord(-4,p)/gcd(ord(-4,p),2). If ord(-4,p) = p-1, then ord(16,p) = (p-1)/2. Conversely, if ord(16,p) = (p-1)/2, then ord(-4,p) = p-1, since otherwise ord(-4,p) = (p-1)/2 is odd, which is impossible since that -4 is not a quadratic residue modulo a prime p == 3 (mod 4).
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Cf. A114564, A302141. A216371 is a supersequence.
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#23 by Jianing Song at Sat Dec 24 05:39:43 EST 2022
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Discussion
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| Sat Dec 24
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| Jianing Song: Why is 1 not included in A163778? By definition, 1 is an odd term in A054639!
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#22 by Jianing Song at Sat Dec 24 05:38:57 EST 2022
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An odd prime p is a term if and only if p == 3 (mod 4), and that the multiplicative order of 2 modulo p is p-1 or (p-1)/2 (p-1 is p == 3 (mod 8), (p-1)/2 if p == 7 (mod 8)).
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proposed
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#21 by Jianing Song at Sat Dec 24 05:36:25 EST 2022
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#20 by Jianing Song at Sat Dec 24 05:36:04 EST 2022
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From Jianing Song, Dec 24 2022: (Start)
Equivalently, primes p == 3 (mod 4) such that the multiplicative order of 4 modulo p is (p-1)/2 (a subsequence of A216371).
Proof of equivalence: Write ord(a,k) be the multiplicative of a modulo k. First we notice that all terms are congruent to 3 modulo 4, since -4 is a quadratic residue modulo p if p == 1 (mod 4). If ord(4,p) = (p-1)/2. Note that (p-1)/2 is odd, so it is coprime to ord(-1,p) = 2. As a result, ord(-4,p) = ((p-1)/2) * 2 = p-1. Conversely, If ord(-4,p) = p-1, we must have ord(4,p) = (p-1)/2 by noting that ord(-4,p) <= lcm(2,ord(4,p)).
It seems that a(n) = 2*A163778(n-1) + 1 for n >= 2. (End)
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Cf. A114564. A216371 is a supersequence.
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approved
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#19 by Jon E. Schoenfield at Tue Jun 13 01:29:54 EDT 2017
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#18 by Jon E. Schoenfield at Tue Jun 13 01:29:49 EDT 2017
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#17 by Jon E. Schoenfield at Tue Jun 13 01:29:46 EDT 2017
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We will use the obvious fact that if a primitive root is a power of an otheranother element, then that other element is also a primitive root. So
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approved
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