A067251 Numbers with no trailing zeros in decimal representation.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104

Offset: 1

Author

Reinhard Zumkeller, Mar 10 2002

Keywords

• nonn
• base
• easy

Comments

Or, decimated numbers: every 10th number has been omitted from the natural numbers. - Cino Hilliard, Feb 21 2005. For example, The 10th number starting with 1 is 10 and is missing from the table because it was decimated.

The word "decimated" can be interpreted in several ways and should be used with caution. - N. J. A. Sloane, Feb 21 2005

Not the same as A052382, as 101 is included.

Numbers in here but not in A043095 are 81, 91, 92, 93, 94,... for example. - R. J. Mathar, Sep 30 2008

The integers 100*a(n) are precisely the numbers whose square ends with exactly 4 identical digits while the integers 10*a(n) form just a subsequence of the numbers whose square ends with exactly 2 identical digits (A346678). - Bernard Schott, Oct 04 2021

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for 10-automatic sequences.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Crossrefs

Complement of A008592.

Cf. A076641 (reversed).

Cf. A004086, A043095, A052382, A168184.

Cf. A039685 (a subsequence), A346678, A346940, A346942.

Programs

Haskell

a067251 n = a067251_list !! (n-1)
a067251_list = filter ((> 0) . flip mod 10) [0..]
-- Reinhard Zumkeller, Jul 11 2015, Dec 29 2011

Maple

S := seq(n + floor((n-1)/9), n=1..100); # Bernard Schott, Oct 04 2021

Mathematica

DeleteCases[Range[110],?(Divisible[#,10]&)] (* _Harvey P. Dale, May 16 2016 *)

PARI

f(n) = for(x=1,n,if(x%10,print1(x","))) \\ Cino Hilliard, Feb 21 2005

PARI

Vec(x*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)/((x-1)^2*(x^2+x+1)*(x^6+x^3+1)) + O(x^100)) \\ Colin Barker, Sep 28 2015

Python

def a(n): return n + (n-1)//9
print([a(n) for n in range(1, 95)]) # Michael S. Branicky, Oct 04 2021

Formula

a(n) = n + floor((n-1)/9).

a(n) mod 10 > 0 for all n.

A004086(A004086(a(n))) = a(n).

A168184(a(n)) = 1. - Reinhard Zumkeller, Nov 30 2009

From Colin Barker, Sep 28 2015: (Start)

a(n) = a(n-1) + a(n-9) - a(n-10) for n>10.

G.f.: x*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1) / ((x-1)^2*(x^2+x+1)*(x^6+x^3+1)). (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (1/20 + 1/sqrt(5) - sqrt(1+2/sqrt(5))/5) * Pi. - Amiram Eldar, May 11 2025

Extensions

Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

Typos corrected in a comment line by Reinhard Zumkeller, Apr 04 2010