A054986 -id:A054986 - cp's OEIS frontend

A055018 Numbers n such that n and n+1 are modest (cf. A054986).

411, 811, 1421, 2036, 2044, 2054, 3054, 4036, 4044, 4054, 8036, 12036, 16036, 20036, 24036, 28036, 32036, 40011, 40044, 50054, 50074, 80011, 88296, 100110, 100270, 100369, 120296, 120404, 140021, 144242, 176296, 200044, 200270, 200369

Offset: 1

Hans Havermann, May 31 2000

			2036 is modest because Mod[2036, 36] = 20. 2037 is modest because Mod[2037, 037] = 2. Hence 2036 is this pair's modest twin representative.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

import Data.List (elemIndices)
a055018 n = a055018_list !! (n-1)
a055018_list = map (a054986 . (+ 1)) $ elemIndices 1 $
               zipWith (-) (tail a054986_list) a054986_list
-- Reinhard Zumkeller, Mar 26 2012

A007627 Primitive modest numbers.

Original entry on oeis.org

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

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

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

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).

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)

Cf. A054986, A055018.

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

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

A210582 Numbers whose first digit is the remainder of their division by the last digit (in base 10).

Original entry on oeis.org

13, 19, 23, 26, 29, 39, 46, 49, 59, 69, 79, 89, 103, 109, 127, 133, 163, 193, 197, 199, 203, 206, 209, 214, 218, 233, 234, 236, 247, 254, 258, 263, 266, 274, 293, 294, 296, 298, 299, 309, 367, 399, 406, 409, 417, 428, 436, 466, 468, 487, 496, 499, 509, 537, 599, 609, 638, 657, 678, 699, 709, 799, 809, 899

Offset: 1

Eric Angelini and M. F. Hasler, Mar 22 2012

This is a restricted or simplified version of the definition of modest numbers A054986.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
E. Angelini, Not Modest, Mar 22 2012
E. Angelini, Not modest [Cached copy, with permission]

Cf. A054986, A178158.

A subsequence of A067251, disjoint with A034709.

a210582 n = a210582_list !! (n-1)
a210582_list = filter (\x -> mod x (a010879 x) == a000030 x) a067251_list
-- Reinhard Zumkeller, Mar 26 2012

[ n: n in [1..1002] | not IsZero(d[1]) and n mod d[1] eq d[#d] where d is Intseq(n) ];  // Bruno Berselli, Mar 26 2012

is_nm( x )=x%10 && x%(x%10)==x\10^(#Str(x)-1)
for(n=1,999,is_nm(n)&print1(n","))

a(n) mod A010879(a(n)) = A000030(a(n)). [Reinhard Zumkeller, Mar 26 2011]

Edited by M. F. Hasler, Jan 14 2014

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A055018 Numbers n such that n and n+1 are modest (cf. A054986).

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A007627 Primitive modest numbers.

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A210582 Numbers whose first digit is the remainder of their division by the last digit (in base 10).

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