A334904 a(n) is the least integer b such that the fractions (b^0)/p, (b^1)/p, ..., (b^(r-1))/p where p is the n-th prime, produce the A006556(n) distinct cycles.

1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 3, 2, 3, 2, 1, 2, 1, 1, 2, 7, 2, 3, 2, 3, 1, 2, 2, 2, 1, 1, 3, 1, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 7, 1, 2, 3, 2, 1, 2, 1, 1, 7, 7, 3, 1, 1, 1, 6, 2, 3, 2, 2, 2, 11, 1, 2, 2, 1, 2, 2, 2, 7, 1, 2, 1, 1, 1, 2, 3, 7, 1, 2, 7, 1, 3, 2, 3, 3, 1, 2, 2, 13

Offset

1,2

Comments

With the exception of the prime numbers 2 and 5, the values of r mentioned above form the sequence A006556.

Detection of all different cycles of digits in the decimal expansions of 1/p, 2/p, ..., (p-1)/p where p=n-th prime. If for the n-th prime p the number of different cycles of digits is equal to r, then there will be the smallest integer b in the interval 0 < b < p with the following property: The fractions (b^0)/p, (b^1)/p, ..., (b^(r-1))/p will produce r different cycles of digits. In this case the term a(n) of the sequence becomes equal to b.

Links

Table of n, a(n) for n=1..92.

Example

For n=13, prime(13)=41, there are A006556(13)=8 cycles.
With b=3, we get (normally, these fractions should be in the form (b^k mod p)/p):
  frac(3^0 / 41) = 0.02439 (1)
  frac(3^1 / 41) = 0.07317 (2)
  frac(3^2 / 41) = 0.21951 (3)
  frac(3^3 / 41) = 0.65853 (4)
  frac(3^4 / 41) = 0.97560 (5)
  frac(3^5 / 41) = 0.92682 (6)
  frac(3^6 / 41) = 0.78048 (7)
  frac(3^7 / 41) = 0.34146 (8=r)
So a(13) = 3.

Prog

(PARI) \\ default(realprecision, 1000)
nbc(p) = (p-1)/znorder(Mod(10, p));
len(p) = znorder(Mod(10, p));
pad(x, sz) = {while(#digits(x) != sz, x*=10); x; }
cmpc(x, y) = {if (x==y, return (0)); my(dx=digits(x), dy=digits(y), v=dx); for (k=1, #dx, v=vector(#v, k, if (k==#v, v[1], v[k+1])); if (v == dy, return (0)); ); return (1); }
decimals(x, sz) = pad(floor(1.0*10^sz*x), sz);
a(n) = {my(p=prime(n)); if ((p==2), return (1)); if ((p==5), return (2)); my(sz=len(p), nb=nbc(p), m=1); while (#vecsort(vector(f(p), k, decimals((m^(k-1) % p)/p, sz)), cmpc, 8) != nb, m++); m; } \\ Michel Marcus, May 29 2020

Crossrefs

Cf. A006556.

Sequence in context: A283468 A100244 A059689 * A165018 A131837 A232628

Adjacent sequences: A334901 A334902 A334903 * A334905 A334906 A334907

Keyword

nonn,base

Author

George Plousos, May 15 2020

Status

approved