A040121 -id:A040121 - cp's OEIS frontend

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

Mario Velucchi (mathchess(AT)velucchi.it)

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

			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)

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

Cf. A087000, A186635.

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

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

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

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

More terms from Reinhard Zumkeller, Jul 29 2003

A040119 Primes p such that x^4 = 10 has a solution mod p.

Original entry on oeis.org

2, 3, 5, 31, 37, 41, 43, 53, 67, 71, 79, 83, 107, 151, 163, 173, 191, 199, 227, 239, 241, 271, 277, 283, 307, 311, 317, 347, 359, 397, 431, 439, 443, 467, 479, 523, 547, 563, 587, 599, 613, 631, 641, 643, 683, 719

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A040121 (complement in the primes).

[p: p in PrimesUpTo(800) | exists(t){x : x in ResidueClassRing(p) | x^4 eq 10}]; // Vincenzo Librandi, Sep 12 2012

ok [p_]:=Reduce[Mod[x^4 - 10, p]== 0, x, Integers]=!= False; Select[Prime[Range[180]], ok] (* Vincenzo Librandi, Sep 12 2012 *)

isA040119(p)={r=0;for(m=0,p-1,if(Mod(m,p)^4==Mod(10,p),r=1));r} \\ Michael B. Porter, Oct 13 2009

select( {is_A040119(p)=ispower(Mod(10, p), 4)}, primes(199)) \\ is_A040119(p) assumes that p is prime, else append "&& isprime(p)". - M. F. Hasler, Nov 19 2024

Definition corrected by Michael B. Porter, Oct 13 2009

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

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A040119 Primes p such that x^4 = 10 has a solution mod p.

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