A037179 -id:A037179 - cp's OEIS frontend

A037178 Longest cycle when squaring modulo n-th prime.

1, 1, 1, 2, 4, 2, 1, 6, 10, 3, 4, 6, 4, 6, 11, 12, 28, 4, 10, 12, 6, 12, 20, 10, 2, 20, 8, 52, 18, 3, 6, 12, 8, 22, 36, 20, 12, 54, 82, 14, 11, 12, 36, 2, 21, 30, 12, 36, 28, 18, 28, 24, 4, 100, 1, 130, 66, 36, 22, 12, 46, 9, 24, 20, 12, 39, 20, 6, 172, 28, 10, 178, 60, 10, 18

Offset: 1

Jud McCranie

nonn

a(n)=1 for Fermat primes, A019434. a(n)=2 for primes in A039687. a(n)=3 for primes in A050527. Sequence A141305 gives those primes p > 3 having the longest possible cycle, (p-3)/2. - T. D. Noe, Jun 24 2008

T. D. Noe, Table of n, a(n) for n=1..10000
E. L. Blanton, Jr., S. P. Hurd and J. S. McCranie, On a digraph defined by squaring modulo n, Fibonacci Quart. 30 (Nov. 1992), 322-333.
Haifeng Xu, The largest cycles consist by the quadratic residues and Fermat primes, arXiv:1601.06509 [math.NT], 2016. See table on page 2.

Cf. A002322, A007733, A256608.
Cf. A019434, A039687, A050527, A141305.
Cf. A037179, A037180.
a(n) = maximal entry in row p of A278185.

a[n_] := Module[{p = Prime[n], k}, k = (p-1)/2^IntegerExponent[p-1, 2]; MultiplicativeOrder[2, k]]; Array[a, 100] (* Jean-François Alcover, Jan 28 2016, after T. D. Noe *)

a(n) = {ppn = prime(n) - 1; k = ppn >> valuation(ppn, 2); znorder(Mod(2, k));} \\ Michel Marcus, Nov 11 2015

rpsi(n) = lcm(znstar(n)[2]); \\ A002322
pb(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ A007733
a(n) = pb(rpsi(prime(n))); \\ Michel Marcus, Jan 28 2016

Let p=prime(n) and k=A000265(p-1), the odd part of p-1. Then a(n) = ord(2,k), that is, the smallest positive integer x such that 2^x = 1 (mod k). - T. D. Noe, Jun 24 2008
a(n) = A007733(A002322(prime(n))). - Michel Marcus, Jan 28 2016
a(n) = A256608(prime(n)).

A037180 Number of different cycle lengths when squaring modulo n-th prime.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 3, 3, 2, 4, 2, 2, 2, 3, 3, 4, 3, 3, 2, 2, 2, 3, 3, 2, 4, 2, 4, 3, 2, 4, 2, 4, 3, 5, 2, 2, 2, 5, 4, 2, 3, 5, 6, 3, 2, 3, 2, 4, 3, 4, 1, 2, 2, 7, 4, 4, 4, 2, 5, 4, 3, 2, 5, 4, 2, 3, 2, 2, 3, 4, 5, 2, 2, 5, 3, 3, 4, 6, 4, 4, 4, 4, 2, 3, 4, 6, 2, 2, 6, 6, 3, 2, 2, 3, 5, 7, 5, 2

Offset: 1

Jud McCranie

nonn

E. L. Blanton, Jr., S. P. Hurd and J. S. McCranie, On a digraph defined by squaring modulo n, Fibonacci Quart. 30 (Nov. 1992), 322-333.

Cf. A037178, A037179.

cp's OEIS Frontend

A037178 Longest cycle when squaring modulo n-th prime.

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