A112927 a(n) is the least prime such that the multiplicative order of 2 mod a(n) equals n, or a(n)=1 if no such prime exists.

1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 23, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 47, 241, 601, 2731, 262657, 29, 233, 331, 2147483647, 65537, 599479, 43691, 71, 37, 223, 174763, 79, 61681, 13367, 5419, 431, 397, 631, 2796203, 2351, 97, 4432676798593, 251, 103, 53, 6361, 87211

Offset: 1

Author

Vladimir Shevelev, Aug 25 2008

Keywords

• nonn

Comments

If a(n) differs from 1, then a(n) is the minimal prime divisor of A064078(n);

a(n)=n+1 iff n+1 is prime from A001122; a(n)=2n+1 iff 2n+1 is prime from A115591.

If a(n) > 1 then a(n) is the index where n occurs first in A014664. - M. F. Hasler, Feb 21 2016

Bang's theorem (special case of Zsigmondy's theorem, see links): a(n)>1 for all n>6. - Jeppe Stig Nielsen, Aug 31 2020

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1206
• Dario Alejandro Alpern, Factorization using the Elliptic Curve Method
• Will Edgington, Factored Mersenne Numbers [from Internet Archive Wayback Machine]
• Wikipedia, Zsigmondy's theorem

Crossrefs

Cf. A002326, A064078, A001122, A115591.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

Programs

PARI

A112927(n,f=factor(2^n-1)[,1])=!for(i=1,#f,znorder(Mod(2,f[i]))==n&&return(f[i])) \\ Use the optional 2nd arg to give a list of pseudoprimes to try when factoring of 2^n-1 is too slow. You may try factor(2^n-1,0)[,1]. - M. F. Hasler, Feb 21 2016