A028416 Primes p such that the decimal expansion of 1/p has a periodic part of even length.

7, 11, 13, 17, 19, 23, 29, 47, 59, 61, 73, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 193, 197, 211, 223, 229, 233, 241, 251, 257, 263, 269, 281, 293, 313, 331, 337, 349, 353, 367, 373, 379, 383, 389, 401, 409, 419, 421, 433

Offset: 1

Author

Mario Velucchi (mathchess(AT)velucchi.it)

Keywords

• nonn
• base

Comments

Primes whose reciprocals have even period length.

Primes p such that the order of 10 mod p is even. - Joerg Arndt, Mar 04 2014

A002371(A049084(a(n))) mod 2 == 0.

Not the same as A040121: a(33)=241 is not in A040121.

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

Examples

			From _Reinhard Zumkeller_, Oct 05 2008: (Start)
(0,5,8,8,2,3,5,2,9,4,1,1,7,6,4,7) is the period of 1/17 (see A007450),
K = A002371(A049084(17))/2 = A002371(7)/2 = 16/2 = 8,
u = 5882352, v = 94117647: u + v = 99999999 = 10^8 - 1. (End)
		

References

• H. Rademacher and O. Toeplitz, Von Zahlen und Figuren (Springer 1930, reprinted 1968), ch. 19, "Die periodischen Dezimalbrueche". [Reinhard Zumkeller, Oct 05 2008]

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to decimal expansion of 1/n.

Crossrefs

Cf. A087000, A186635.

Programs

Maple

A028416 := proc(n) local st:
st := ithprime(n):
if (modp(numtheory[order](10,st),2) = 0) then
   RETURN(st)
fi: end:  seq(A028416(n), n=1..100); # Jani Melik, Feb 24 2011

Mathematica

Select[Prime[Range[4,100]],EvenQ[Length[RealDigits[1/#][[1,1]]]]&] (* Harvey P. Dale, Jul 07 2011 *)

PARI

forprime(p=7,1e3,if(znorder(Mod(10,p))%2==0,print1(p", "))) \\ Charles R Greathouse IV, Feb 24 2011

Python

from sympy import gcd, isprime, n_order
is_A028416 = lambda n: gcd(n,10)==1 and n>5 and n_order(10, n)%2==0 and isprime(n) # M. F. Hasler, Nov 19 2024

Extensions

More terms from Reinhard Zumkeller, Jul 29 2003