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

1,2

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

Links

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Index entries for 10-automatic sequences.

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)

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A284990 A238621 A099102 * A387225 A366728 A256964

Adjacent sequences: A001740 A001741 A001742 * A001744 A001745 A001746

Keyword

base,nonn,easy

Author

N. J. A. Sloane

Extensions

Examples added by Hieronymus Fischer, May 30 2012

Status

approved