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 A320383 Multiplicative order of 3/2 modulo n-th prime. 2
 2, 6, 10, 4, 16, 3, 11, 7, 30, 36, 40, 21, 23, 13, 58, 12, 33, 7, 36, 26, 82, 88, 8, 25, 102, 106, 108, 112, 126, 130, 136, 69, 74, 150, 156, 81, 83, 86, 178, 36, 95, 96, 49, 66, 5, 222, 226, 228, 232, 119, 30, 250, 256, 131, 67, 270, 276, 40, 141, 73, 51, 155, 156, 79, 11, 168, 346, 348, 352, 179, 366, 124 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS Let p = prime(n). a(n) is the smallest positive k such that p divides 3^k - 2^k. Obviously, a(n) divides p - 1. If a(n) = p - 1, then p is listed in A320384. If p == 1, 5, 19, 23 (mod 24), then 3/2 is a quadratic residue modulo p, so a(n) divides (p - 1)/2. By Zsigmondy's theorem, for each k >=2 there is a prime that divides 3^k-2^k but not 3^j-2^j for j < k.  Therefore each integer >= 2 appears in the sequence at least once. - Robert Israel, Apr 20 2021 LINKS Robert Israel, Table of n, a(n) for n = 3..10000 Wikipedia, Multiplicative order Wikipedia, Zsigmondy's theorem EXAMPLE Let ord(n,p) be the multiplicative order of n modulo p. 3/2 == 4 (mod 5), so a(3) = ord(4,5) = 2. 3/2 == 5 (mod 7), so a(4) = ord(5,7) = 6. 3/2 == 7 (mod 11), so a(5) = ord(7,11) = 10. 3/2 == 8 (mod 13), so a(6) = ord(8,13) = 4. MAPLE f:= proc(n) local p; p:= ithprime(n); numtheory:-order(3/2 mod p, p) end proc: map(f, [\$3..100]); # Robert Israel, Apr 20 2021 PROG (PARI) forprime(p=5, 10^3, print1(znorder(Mod(3/2, p)), ", ")) \\ Joerg Arndt, Oct 13 2018 CROSSREFS Cf. A001047, A211242, A320384. Sequence in context: A095105 A220338 A052194 * A073662 A086553 A342068 Adjacent sequences:  A320380 A320381 A320382 * A320384 A320385 A320386 KEYWORD nonn,look AUTHOR Jianing Song, Oct 12 2018 STATUS approved

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Last modified May 15 10:54 EDT 2021. Contains 343909 sequences. (Running on oeis4.)