A072982 Primes p for which the period of 1/p is a power of 2.

3, 11, 17, 73, 101, 137, 257, 353, 449, 641, 1409, 10753, 15361, 19841, 65537, 69857, 453377, 976193, 1514497, 5767169, 5882353, 6187457, 8253953, 8257537, 70254593, 167772161, 175636481, 302078977, 458924033, 639631361, 1265011073

Offset: 1

Author

Benoit Cloitre, Jul 26 2002

Keywords

• nonn
• base

Comments

All Fermat primes > 5 (A019434) are in the sequence, since it can be shown that the period of 1/(2^(2^n)+1) is 2^(2^n) whenever 2^(2^n)+1 is prime. - Benoit Cloitre, Jun 13 2007

Take all the terms from row 2^k of triangle in A046107 for k >= 0 and sort to arrive at this sequence. - Ray Chandler, Nov 04 2011

Additional terms, but not necessarily the next in sequence: 13462517317633 has period 1048576 = 2^20; 46179488366593 has period 2199023255552 = 2^41; 101702694862849 has period 8388608 = 2^23; 171523813933057 has period 4398046511104 = 2^42; 505775348776961 has period 2199023255552 = 2^41; 834427406578561 has period 64 = 2^6 - Ray Chandler, Nov 09 2011

Furthermore (excluding the initial term 3) this sequence is also the ascending sequence of primes dividing 10^(2^k)+1 for some nonnegative integer k. For a prime dividing 10^(2^k)+1, the period of 1/p is 2^(k+1). Thus for the prime p = 558711876337536212257947750090161313464308422534640474631571587847325442162307811\

65223702155223678309562822667655169, a factor of 10^(2^7)+1, the period of 1/p is only 2^8. This large prime then also belongs to the sequence. - Christopher J. Smyth, Mar 13 2014

For any m, every term that is not a factor of 10^(2^k)-1 for some k < m is congruent to 1 (mod 2^m). Thus all terms except 3, 11, 17, 73, 101, 137, 353, 449, 69857, 976193, 5882353, 6187457 are congruent to 1 (mod 128). - Robert Israel, Jun 17 2016

Additional terms listed earlier confirmed as next terms in sequence. - Arkadiusz Wesolowski, Jun 17 2016

Examples

			15361 has a period of 256 = 2^8, hence 15361 is in the sequence.
		

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 1..45 (first 33 terms from Ray Chandler, to 36 terms from Robert G. Wilson v, to 39 terms from Ray Chandler)
• Ray Chandler, Known Terms of A072982
• Wilfrid Keller, Prime factors of generalized Fermat numbers Fm(10) and complete factoring status
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Cf. A002371, A007138, A046107, A054471.

Cf. A197224 (power of 2 which is the period of the decimal 1/a(n)).

Programs

Maple

filter:= proc(p) local k;
  if not isprime(p) then return false fi;
  k:=igcd(p-1,2^ilog2(p));
  evalb(10 &^ k mod p = 1)
end proc:
r:= select(`<=`,`union`(seq(numtheory:-factorset(10^(2^k)-1),k=1..6)),10^9):
b:= select(filter, {seq(i,i=129..10^9,128)}):
sort(convert(r union b, list)); # Robert Israel, Jun 17 2016

Mathematica

Do[ If[ IntegerQ[ Log[2, Length[ RealDigits[ 1/Prime[n]] [[1, 1]]]]], Print[ Prime[n]]], {n, 1, 47500}] (* Robert G. Wilson v, May 09 2007 *)
pmax = 10^10; p = 1; While[p < pmax,p = NextPrime[p];If[ IntegerQ[Log[2, MultiplicativeOrder[10, p] ] ], Print[ p];];]; (* Ray Chandler, May 14 2007 *)

PARI

select( {is_A072982(p)=if(p>5, 1<M. F. Hasler, Nov 18 2024

Python

from itertools import count, islice
from sympy import prime, n_order
def A072982_gen(): return (p for p in (prime(n) for n in count(2)) if p != 5 and bin(n_order(10,p))[2:].rstrip('0') == '1')
A072982_list = list(islice(A072982_gen(),10)) # Chai Wah Wu, Feb 07 2022

Python

from sympy import primerange, n_order
A072982_upto = lambda N=1e5: [p for p in primerange(3, N) if p != 5 and n_order(10, p).bit_count() == 1] # or (...) to get a generator. - M. F. Hasler, Nov 19 2024

Extensions

Edited by Robert G. Wilson v, Aug 20 2002

a(18) from Ray Chandler, May 02 2007

a(19) from Robert G. Wilson v, May 09 2007

a(20)-a(32) from Ray Chandler, May 14 2007

Deleted an unsatisfactory PARI program. - N. J. A. Sloane, Nov 19 2024