

A072982


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


12



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
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OFFSET

1,1


COMMENTS

All Fermat primes>5 (A019434) are in the sequence, since it can be shown that the period length 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 length 1048576 = 2^20; 46179488366593 has period length 2199023255552 = 2^41; 101702694862849 has period length 8388608 = 2^23; 171523813933057 has period length 4398046511104 = 2^42; 505775348776961 has period length 2199023255552 = 2^41; 834427406578561 has period length 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 length 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


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


EXAMPLE

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


MAPLE

filter:= proc(p) local k;
if not isprime(p) then return false fi;
k:=igcd(p1, 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 *)
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 *)


PROG

(PARI) ? a(n)=if(n<4, n==2, znorder(Mod(10, prime(n)))) ? for(n=1, 20000, if(gcd(a(n), 2^1000)==a(n), print1(prime(n), ", ")))


CROSSREFS

Cf. A002371, A007138, A046107, A054471.
Cf. A197224 (power of 2 which is the period of the decimal 1/a(n)).
Sequence in context: A032008 A061368 A145701 * A124787 A080306 A036448
Adjacent sequences: A072979 A072980 A072981 * A072983 A072984 A072985


KEYWORD

nonn,base


AUTHOR

Benoit Cloitre, Jul 26 2002


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


STATUS

approved



