A186635 Primes p such that the decimal expansion of 1/p has a periodic part of odd length.

2, 3, 5, 31, 37, 41, 43, 53, 67, 71, 79, 83, 107, 151, 163, 173, 191, 199, 227, 239, 271, 277, 283, 307, 311, 317, 347, 359, 397, 431, 439, 443, 467, 479, 523, 547, 563, 587, 599, 613, 631, 643, 683, 719, 733, 751, 757, 773, 787, 797, 827, 839, 853, 883, 907, 911, 919, 947, 991, 1013, 1031, 1039, 1093, 1123, 1151, 1163, 1187

Offset: 1

Author

Jani Melik, Feb 24 2011

Keywords

• nonn
• base

Comments

Interestingly, the initial terms of A040119 (Primes p such that x^4 = 10 has a solution mod p) are identical to the initial terms of this sequence except for 241 which is a term of A040119 but not of A186635. [John W. Layman, Feb 25 2011]

There are many numbers in A040119 that are not here: 241, 641, 769, 809, 1009, 1409, 1601, 1721.... - T. D. Noe, Feb 25 2011

Crossrefs

Cf. A002371, A048595, A028416 (complement in the primes), A040119.

Programs

Maple

Ax := proc(n) local st:
st := ithprime(n):
if (modp(numtheory[order](10,st),2) <> 0) then
   RETURN(st)
fi: end:  seq(Ax(n), n=1..200);

Mathematica

Union[{2, 5}, Select[Prime[Range[200]], OddQ[Length[RealDigits[1/#][[1, 1]]]] &]]

PARI

select( {is_A186635(n)=isprime(n) && (n<7 || znorder(Mod(10, n))%2)}, [0..1234]) \\ M. F. Hasler, Nov 19 2024

Python

from sympy import isprime, n_order
is_A186635 = lambda n: isprime(n) and (n<7 or n_order(10, n)%2)
[n for n in range(1234) if is_A186635(n)] # M. F. Hasler, Nov 19 2024