%I #49 Jun 13 2022 03:03:13
%S 7,11,13,17,19,23,29,47,59,61,73,89,97,101,103,109,113,127,131,137,
%T 139,149,157,167,179,181,193,197,211,223,229,233,241,251,257,263,269,
%U 281,293,313,331,337,349,353,367,373,379,383,389,401,409,419,421,433
%N Primes p such that the decimal expansion of 1/p has a periodic part of even length.
%C Primes whose reciprocals have even period length.
%C Primes p such that the order of 10 mod p is even. - _Joerg Arndt_, Mar 04 2014
%C A002371(A049084(a(n))) mod 2 == 0.
%C Not the same as A040121: a(33)=241 is not in A040121.
%C Let (d(i): 1<=i<=2*K) be the period of the decimal expansion of 1/a(n), K=A002371(A049084(a(n)))/2, then d(i) + d(i+K) = 9 for i with 1<=i<=K, or, equivalently: u + v = 10^K - 1 with u = Sum_{i=1..K} d(i)*10^(K-i) and v = Sum_{i=1..K} d(i+K)*10^(K-i). - _Reinhard Zumkeller_, Oct 05 2008
%D H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, "Die periodischen Dezimalbrueche". [_Reinhard Zumkeller_, Oct 05 2008]
%H T. D. Noe, <a href="/A028416/b028416.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>.
%e From _Reinhard Zumkeller_, Oct 05 2008: (Start)
%e (0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7) is the period of 1/17 (see A007450),
%e K = A002371(A049084(17))/2 = A002371(7)/2 = 16/2 = 8,
%e u = 5882352, v = 94117647: u + v = 99999999 = 10^8 - 1. (End)
%p A028416 := proc(n) local st:
%p st := ithprime(n):
%p if (modp(numtheory[order](10,st),2) = 0) then
%p RETURN(st)
%p fi: end: seq(A028416(n), n=1..100); # _Jani Melik_, Feb 24 2011
%t Select[Prime[Range[4,100]],EvenQ[Length[RealDigits[1/#][[1,1]]]]&] (* _Harvey P. Dale_, Jul 07 2011 *)
%o (PARI) forprime(p=7,1e3,if(znorder(Mod(10,p))%2==0,print1(p", "))) \\ _Charles R Greathouse IV_, Feb 24 2011
%Y Cf. A087000, A186635.
%K nonn,base
%O 1,1
%A Mario Velucchi (mathchess(AT)velucchi.it)
%E More terms from _Reinhard Zumkeller_, Jul 29 2003