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A007627 Primitive modest numbers.

%I A007627 M4888 #32 Dec 20 2017 23:29:39
%S A007627 13,19,23,29,49,59,79,89,103,109,111,133,199,203,209,211,233,299,311,
%T A007627 409,411,433,499,509,511,533,599,611,709,711,733,799,809,811,833,899,
%U A007627 911,1003,1009,1011,1027,1033,1037,1099,1111
%N A007627 Primitive modest numbers.
%C A007627 Modest numbers (A054986) are the same but without assuming (a,b) = 1.
%C A007627 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),
%C A007627 m = d(10^k - 1) the number of divisors of n (A000005),
%C A007627 and phi is Euler totient function (A000010).
%C A007627 E.g., for k = 1: b = 1, 3, 9, and pairs of (a,b) are:
%C A007627 (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
%D A007627 Problem 1291, J. Rec. Math., 17 (No.2, 1984), 140-141.
%D A007627 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007627 Reinhard Zumkeller, <a href="/A007627/b007627.txt">Table of n, a(n) for n = 1..10000</a>
%H A007627 H. Havermann, <a href="/A007627/a007627.pdf">Modest numbers</a>, J. Recreational Mathematics, 17.2 (1984), 140-141. (Annotated scanned copy)
%F A007627 n = a*10^k + b such that (a, b)=1, n == a (mod b), a < b < 10^k.
%o A007627 (Haskell)
%o A007627 import Data.List (inits, tails)
%o A007627 a007627 n = a007627_list !! (n-1)
%o A007627 a007627_list = filter modest' [1..] where
%o A007627    modest' x = or $ zipWith m
%o A007627                (map read $ (init $ tail $ inits $ show x) :: [Integer])
%o A007627                (map read $ (tail $ init $ tails $ show x) :: [Integer])
%o A007627       where m u v = u < v && (x - u) `mod` v == 0 && gcd u v == 1
%o A007627 -- _Reinhard Zumkeller_, Mar 27 2011
%Y A007627 Cf. A054986, A055018.
%K A007627 nonn,easy,base
%O A007627 1,1
%A A007627 _N. J. A. Sloane_, _Robert G. Wilson v_, _Mira Bernstein_