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

1,2

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

Prog

(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

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).

Sequence in context: A112071 A231609 A046561 * A097406 A064078 A292015

Adjacent sequences: A112924 A112925 A112926 * A112928 A112929 A112930

Keyword

nonn,changed

Author

Vladimir Shevelev, Aug 25 2008

Status

approved