A097443 - OEIS

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

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OFFSET

1,1

COMMENTS

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

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

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Almost the same as A001914.

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

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

Sequence in context: A235265 A347988 A275081 * A248368 A171517 A179026

Adjacent sequences: A097440 A097441 A097442 * A097444 A097445 A097446

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved