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A028416 Primes p such that the decimal expansion of 1/p has a periodic part of even length. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
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
EXAMPLE
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)
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 *)
PROG
(PARI) forprime(p=7, 1e3, if(znorder(Mod(10, p))%2==0, print1(p", "))) \\ Charles R Greathouse IV, Feb 24 2011
CROSSREFS
Sequence in context: A135776 A067831 A086998 * A040121 A156114 A304690
KEYWORD
nonn,base
AUTHOR
Mario Velucchi (mathchess(AT)velucchi.it)
EXTENSIONS
More terms from Reinhard Zumkeller, Jul 29 2003
STATUS
approved

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Last modified February 27 10:59 EST 2024. Contains 370378 sequences. (Running on oeis4.)