A007627 Primitive modest numbers.

13, 19, 23, 29, 49, 59, 79, 89, 103, 109, 111, 133, 199, 203, 209, 211, 233, 299, 311, 409, 411, 433, 499, 509, 511, 533, 599, 611, 709, 711, 733, 799, 809, 811, 833, 899, 911, 1003, 1009, 1011, 1027, 1033, 1037, 1099, 1111

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Keywords

• nonn
• easy
• base

Comments

Modest numbers (A054986) are the same but without assuming (a,b) = 1.

For given k, (see FORMULA section) b's are divisors of (10^k - 1), and a's are coprime with b, number of pairs (a,b) is Sum_{i=1..m} phi(b_i) - 1 where b_i are divisors of (10^k - 1),

m = d(10^k - 1) the number of divisors of n (A000005),

and phi is Euler totient function (A000010).

E.g., for k = 1: b = 1, 3, 9, and pairs of (a,b) are:

(1,3), (2,3), (1,9), (2,9), (4,9), (5,9), (7,9), and (8,9) - a total of 8 pairs. - Zak Seidov, Mar 22 2012

References

• Problem 1291, J. Rec. Math., 17 (No.2, 1984), 140-141.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• H. Havermann, Modest numbers, J. Recreational Mathematics, 17.2 (1984), 140-141. (Annotated scanned copy)

Crossrefs

Cf. A054986, A055018.

Programs

Haskell

import Data.List (inits, tails)
a007627 n = a007627_list !! (n-1)
a007627_list = filter modest' [1..] where
   modest' x = or $ zipWith m
               (map read $ (init $ tail $ inits $ show x) :: [Integer])
               (map read $ (tail $ init $ tails $ show x) :: [Integer])
      where m u v = u < v && (x - u) `mod` v == 0 && gcd u v == 1
-- Reinhard Zumkeller, Mar 27 2011

Formula

n = a*10^k + b such that (a, b)=1, n == a (mod b), a < b < 10^k.