A001743 Numbers in which every digit contains at least one loop (version 1).

0, 6, 8, 9, 60, 66, 68, 69, 80, 86, 88, 89, 90, 96, 98, 99, 600, 606, 608, 609, 660, 666, 668, 669, 680, 686, 688, 689, 690, 696, 698, 699, 800, 806, 808, 809, 860, 866, 868, 869, 880, 886, 888, 889, 890, 896, 898, 899, 900, 906, 908, 909, 960, 966, 968, 969

Offset: 1

Author

N. J. A. Sloane

Keywords

• base
• nonn
• easy

Comments

See A001744 for the other version.

If n-1 is represented as a base-4 number (see A007090) according to n-1 = d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n) = Sum_{j=0..m} c(d(j))*10^j, where c(k)=0,6,8,9 for k=0..3. - Hieronymus Fischer, May 30 2012

Examples

			a(1000) = 99896.
a(10^4) = 8690099.
a(10^5) = 680688699.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..10000
• Index entries for 10-automatic sequences.

Crossrefs

Cf. A007090, A046034, A029581, A084984, A017042, A001744, A014261, A014263, A202267, A202268.

Programs

Mathematica

Union[Flatten[Table[FromDigits/@Tuples[{0,6,8,9},n],{n,3}]]] (* Harvey P. Dale, Sep 04 2013 *)

PARI

is(n) = #setintersect(vecsort(digits(n), , 8), [1, 2, 3, 4, 5, 7])==0 \\ Felix Fröhlich, Sep 09 2019

Formula

From Hieronymus Fischer, May 30 2012: (Start)

a(n) = ((b_m(n)+6) mod 9 + floor((b_m(n)+2)/3) - floor(b_m(n)/3))*10^m + Sum_{j=0..m-1} (b_j(n) mod 4 +5*floor((b_j(n)+3)/4) +floor((b_j(n)+2)/4)- 6*floor(b_j(n)/4)))*10^j, where n>1, b_j(n)) = floor((n-1-4^m)/4^j), m = floor(log_4(n-1)).

a(1*4^n+1) = 6*10^n.

a(2*4^n+1) = 8*10^n.

a(3*4^n+1) = 9*10^n.

a(n) = 6*10^log_4(n-1) for n=4^k+1,

a(n) < 6*10^log_4(n-1), otherwise.

a(n) > 10^log_4(n-1) for n>1.

a(n) = 6*A007090(n-1), iff the digits of A007090(n-1) are 0 or 1.

G.f.: g(x) = (x/(1-x))*Sum_{j>=0} 10^j*x^4^j *(1-x^4^j)* (6 + 8x^4^j + 9(x^2)^4^j)/(1-x^4^(j+1)).

Also: g(x) = (x/(1-x))*(6*h_(4,1)(x) + 2*h_(4,2)(x) + h_(4,3)(x) - 9*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*(x^4^j)^k/(1-(x^4^j)^4). (End)

Extensions

Examples added by Hieronymus Fischer, May 30 2012