A069570 Numbers n in which the k-th digit (counted from the right) is nonzero and either a divisor or a multiple of k, for all 1 <= k <= number of digits of n.

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, 41, 42, 43, 44, 45, 46, 47, 48, 49, 61, 62, 63, 64, 65, 66, 67, 68, 69, 81, 82, 83, 84, 85, 86, 87, 88, 89, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125

Offset: 1

Author

Amarnath Murthy, Mar 24 2002

Keywords

• nonn
• base

Comments

The units digits are 1, ..., 9 repeating (period 9). From n = 10 on, the 10's digits are { 1, 2, 4, 6, 8 } each repeated 9 times, and then starting over with 1. Similarly, starting with the first 3-digit term, the 100's digits are {1, 3, 6, 9}, each repeated 45 times, then starting over with 1. From the first 4-digit term on, the 1000's digits are { 1, 2, 4, 8 }, each repeated 180 times, then starting over with 1, etc. - M. F. Hasler, Sep 27 2016

Examples

			The only restriction on the units digit is that it is nonzero. Therefore all single-digit numbers are included.
23 is a term because the 1st digit from the right is 3 which is a multiple of 1, and the 2nd digit from the right is 2 which is a multiple and also divisor of 2.
More generally, the second digit from the right ("10s digit") must be 1 or even.
Similarly, the third digit from the right must be 1, 3 6 or 9.
As all repunits are in the sequence, the sequence is infinite.
		

Crossrefs

Cf. A069571.

Programs

Mathematica

Select[Range@ 125, Times @@ Map[Boole, MapIndexed[If[#1 == 0, False, Total@ Boole@ {First@ Divisible[#2, #1], First@ Divisible[#1, #2]} > 0] &, Reverse@ IntegerDigits@ #]] > 0 &] (* Michael De Vlieger, Sep 27 2016 *)

PARI

select( is(n)=!for(i=1,#n=Vecrev(digits(n)),(!n[i]||(n[i]%i&&i%n[i]))&&return), [1..125]) \\ M. F. Hasler, Sep 27 2016

PARI

is(n) = {my(d = digits(n)); for(i=1,#d, m=min(d[#d+1-i], i); if(m==0, return(0)); if((d[#d+1-i] + i)%m!=0,return(0)));1} \\ David A. Corneth, Sep 27 2016

PARI

A069570(n,s,k,d)={until(!n\=#d,s+=10^(k++-1)*(d=select(d->!(k%d&&d%k),[1..9]))[n--%#d+1]);s} \\ M. F. Hasler, Sep 28 2016

Formula

a(n) % 10 = (n-1) % 9 + 1. - M. F. Hasler, Sep 27 2016

Extensions

Corrected (inserted missing terms) and extended by Jeremy Gardiner, Jun 17 2010

Definition clarified by M. F. Hasler, Sep 27 2016