A007627 Primitive modest numbers. (Formerly M4888)

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

1,1

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)

Formula

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

Prog

(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

Crossrefs

Cf. A054986, A055018.

Sequence in context: A191020 A180545 A113017 * A180525 A257590 A121877

Adjacent sequences: A007624 A007625 A007626 * A007628 A007629 A007630

Keyword

nonn,easy,base

Author

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

Status

approved