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

1,1

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

Equivalently, the set of primes p such that 10^n + 1 is never divisible by p for any n. This is exactly 2 and 5 plus the primes p such that the multiplicative order of p modulo n is odd, which is this sequence. - Ely Golden, Dec 09 2025

Links

Table of n, a(n) for n=1..67.

Brady Haran and Matt Parker, 100000001 is divisible by 17, Numberphile video (2025).

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]]]] &]]

Prog

(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

Crossrefs

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

Sequence in context: A090475 A060301 A040119 * A265807 A106308 A036797

Adjacent sequences: A186632 A186633 A186634 * A186636 A186637 A186638

Keyword

nonn,base

Author

Jani Melik, Feb 24 2011

Status

approved