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A226181 Primes p such that p-1 divided by the period of the binary expansion of 1/p equals 2^x for some nonnegative integer x. 3
3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 227, 233, 239, 257, 263, 269, 271, 281, 293, 311, 313, 317, 337, 347, 349 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Equivalently, p-1 divided by the period of the decimal expansion of 1/p equals 2^x for some nonnegative integer x. Composite numbers satisfying this condition are given in A243050. - Lear Young, May 30 2013

Let pi_1(x) and pi(x) be the numbers of primes of this sequence and all primes not exceeding x, respectively. Then, for x>=3, p_1(x)/pi(x) >= C_Artin = 0.37395581... Numerical results suggest that it is likely lim pi_1(x)/pi(x) = 2*C_Artin. - Peter J. C. Moses and Vladimir Shevelev, May 29 2014

LINKS

Lear Young, Table of n, a(n) for n = 1..1000

EXAMPLE

(41-1)/20 = 2. 20 is the period of the binary representation of 1/n, the odd part of 2 is 1.

MATHEMATICA

Select[Prime[Range[2, 100]], # == 2^IntegerExponent[#, 2] &[(# - 1)/MultiplicativeOrder[2, #]] &] (* Peter J. C. Moses, May 28 2014 *)

PROG

(PARI)

is(n) = {

  m = valuation(n+1, 2);

      k=(n+1)>>m;

      if(k!=1, for(i=0, (n-1)>>1,

        l=valuation(k+n, 2);

        k=(k+n)>>l;

        m+=l; if(k==1, break)));

       ((n-1)/m)>>valuation((n-1)/m, 2)==1

       \\ m  equals znorder(Mod(2, n))

    }

forstep(i=3, 1e3, 2, if(is(i), print1(i, ", ")))

\\ Lear Young May 30 2013

(PARI)

forstep(i=1, 1e3, 2, j = (i-1)/znorder(Mod(2, i)); if(j>>valuation(j, 2)==1, print1(i, ", "))) \\ Lear Young May 31 2013

CROSSREFS

Cf. A007733, A136042.

Sequence in context: A002556 A130101 A130057 * A120637 A278454 A064534

Adjacent sequences:  A226178 A226179 A226180 * A226182 A226183 A226184

KEYWORD

nonn,easy

AUTHOR

Lear Young, May 30 2013

STATUS

approved

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Last modified March 18 16:00 EDT 2019. Contains 321292 sequences. (Running on oeis4.)