A097443 Half-period primes, i.e., primes p for which the decimal expansion of 1/p has period (p-1)/2.

3, 13, 31, 43, 67, 71, 83, 89, 107, 151, 157, 163, 191, 197, 199, 227, 283, 293, 307, 311, 347, 359, 373, 401, 409, 431, 439, 443, 467, 479, 523, 557, 563, 569, 587, 599, 601, 631, 653, 677, 683, 719, 761, 787, 827, 839, 877, 881, 883, 911, 919, 929, 947, 991

Offset: 1

Author

Julien Peter Benney (jpbenney(AT)ftml.net), Aug 23 2004

Keywords

• nonn

Comments

Primes p for which 10 has multiplicative order (p-1)/2. - Robert Israel, Jul 15 2016

Examples

			13 is a half-period prime because 1/13 = 0.076923076923076923076923..., which has period 6, or (13-1)/2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (Terms 2..1001 from T. D. Noe.)
• Makoto Kamada, Factorizations of 11...11 (Repunit).
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Almost the same as A001914.

Cf. A001913, A055628, A056157, A056210-A056217, A098680

Cf. also A000040, A007732, A006883, A097443, A097955.

Programs

Maple

select(t -> isprime(t) and numtheory:-order(10, t) = (t-1)/2,
[seq(t,t = 3..1000,2)]); # Robert Israel, Jul 15 2016

Mathematica

f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 200]], f[ # ] == 2 &] (* Robert G. Wilson v, Sep 14 2004 *)

PARI

is(n)= gcd(10,n)==1 && isprime(n) && znorder(Mod(10,n))==(n-1)/2 \\ Dana Jacobsen, Jul 19 2016

Perl

use ntheory ":all"; forprimes { say if znorder(10,$) == ($-1)/2; } 1,1000; # Dana Jacobsen, Jul 19 2016

Extensions

Edited (including prepending 3), at the suggestion of Georg Fischer, by N. J. A. Sloane, Oct 19 2018