A060370 Ratios (p-1)/d, where p is a prime and d is the number of digits of the periodic part of the decimal expansion of 1/p.

1, 2, 4, 1, 5, 2, 1, 1, 1, 1, 2, 12, 8, 2, 1, 4, 1, 1, 2, 2, 9, 6, 2, 2, 1, 25, 3, 2, 1, 1, 3, 1, 17, 3, 1, 2, 2, 2, 1, 4, 1, 1, 2, 1, 2, 2, 7, 1, 2, 1, 1, 34, 8, 5, 1, 1, 1, 54, 4, 10, 2, 2, 2, 2, 1, 4, 3, 1, 2, 3, 11, 2, 1, 2, 1, 1, 1, 4, 2, 2, 1, 3, 2, 1, 2

Offset: 1

Author

Klaus Brockhaus, Apr 01 2001

Keywords

• nonn
• base

Comments

The sequence of 2nd, 4th and following terms coincides with A006556, which gives the "number of different cycles of digits in the decimal expansions of 1/p, 2/p, ..., (p-1)/p where p = n-th prime different from 2 or 5".

Examples

			a(13) = 40/5 = 8, since 41 is the 13th prime and the periodic part of 1/41 = 0.02439024390... consists of 5 digits.
		

Links

• Carmine Suriano and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 101 terms from Carmine Suriano)

Crossrefs

Cf. A000040, A060283, A048595, A006556.

Programs

Mathematica

Join[{1, 2, 4}, Table[p = Prime[n]; (p - 1)/Length[RealDigits[1/p, 10][[1, 1]]], {n, 4, 100}]] (* T. D. Noe, Oct 04 2012 *)

Python

from sympy import prime, n_order
def A060370(n): return 1 if n == 1 or n == 3 else n_order(10, prime(n))
print([(prime(n)-1)//A060370(n) for n in range(1,86)]) # Karl-Heinz Hofmann, Mar 16 2022

Formula

a(n) = (b(n)-1)/c(n), where b(n) and c(n) are the n-th terms of A000040 and A048595 respectively.