login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A037178 Longest cycle when squaring modulo n-th prime. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

LINKS

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.

FORMULA

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

MATHEMATICA

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

PROG

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

(PARI) 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

CROSSREFS

Cf. A019434, A039687, A050527, A141305.

Cf. A037179, A037180.

Sequence in context: A021417 A105791 A116515 * A077748 A152753 A113973

Adjacent sequences:  A037175 A037176 A037177 * A037179 A037180 A037181

KEYWORD

nonn

AUTHOR

Jud McCranie

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 6 03:02 EST 2016. Contains 278771 sequences.