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 A322906 The number of zeros in the fundamental Pisano period of the 3-Fibonacci numbers A006190 modulo n. 11
 1, 1, 1, 1, 4, 1, 2, 2, 1, 4, 2, 1, 4, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 4, 4, 1, 2, 4, 2, 2, 2, 2, 2, 2, 1, 4, 2, 2, 2, 4, 2, 1, 2, 2, 1, 2, 2, 2, 4, 2, 2, 1, 1, 2, 2, 2, 4, 2, 2, 1, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 4, 4, 2, 2, 2, 2, 1, 2, 1, 4, 2, 2, 2, 1, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS a(n) is the multiplicative order of A006190(A322907(n)+1) modulo n. a(n) has value 1, 2 or 4. This is because A006190(k,m+1)^4 == 1 (mod A006190(k,m)). Conjecture: For primes p == 1, 9, 17, 25, 49, 81 (mod 104), the probability of a(p^e) taking on the value 1, 2, 4 is 1/6, 2/3, 1/6, respectively; for primes p == 29, 53, 61, 69, 77, 101 (mod 104), the probability of a(p^e) taking on the value 1, 4 is 1/2, 1/2, respectively. LINKS Jianing Song, Table of n, a(n) for n = 1..1000 FORMULA For n > 2, T(n,k) = 4 iff A322907(n) is odd; 1 iff A322907(n) is even but not divisible by 4; 2 iff A322907(n) is divisible by 4. For primes p == 3, 23, 27, 35, 43, 51 (mod 52), a(p^e) = 1. For primes p == 5, 21, 33, 37, 41, 45 (mod 52), a(p^e) = 4. For primes p == 7, 11, 15, 19, 31, 47 (mod 52), a(p^e) = 2. a(13^e) = 4. a(2^e) = 1 if e = 1, 2 and 2 if e >= 3. a(n) = A175182(n)/A322907(n). PROG (PARI) A006190(m) = ([3, 1; 1, 0]^m)[2, 1] a(n) = my(i=1); while(A006190(i)%n!=0, i++); znorder(Mod(A006190(i+1), n)) CROSSREFS Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = k*x(n+1) + x(n). Then the periods, ranks and the ratios of the periods to the ranks modulo a given integer n are given by: k = 1: A001175 (periods), A001177 (ranks), A001176 (ratios). k = 2: A175181 (periods), A214028 (ranks), A214027 (ratios). k = 3: A175182 (periods), A322907 (ranks), this sequence (ratios). Sequence in context: A300657 A112621 A081448 * A106437 A279605 A054713 Adjacent sequences:  A322903 A322904 A322905 * A322907 A322908 A322909 KEYWORD nonn AUTHOR Jianing Song, Jan 05 2019 STATUS approved

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Last modified November 26 03:20 EST 2020. Contains 338632 sequences. (Running on oeis4.)