A113705 Inverse Moebius transform of powers of 10.

1, 10, 110, 1010, 10110, 100010, 1001110, 10000010, 100010110, 1000001010, 10000100110, 100000000010, 1000001011110, 10000000000010, 100000010000110, 1000000000101010, 10000000100010110, 100000000000000010, 1000000001001001110, 10000000000000000010, 100000000010000110110

Offset: 0

Author

Paul Barry, Nov 05 2005

Keywords

• easy
• nonn

Comments

A055895 to base 2.

The triangle formed by stacking the reversals of a(n) is A113704.

Using decimal positional notation, write a 1 at position d for all divisors d of n. All other digits are zeros and leading zeros are not permitted. - Michael De Vlieger, May 24 2017

Examples

			From _Michael De Vlieger_, May 24 2017: (Start)
First 20 terms of a(n), replacing zeros with "." to more clearly show positions of 1s in positions corresponding to terms in row n of A027750. This chart also pertains to terms of A055895 written in binary.
   n                    a(n)  A027750(n)
   ---------------------------------------
   0:                      1  {}
   1:                     1.  1
   2:                    11.  1,2
   3:                   1.1.  1,3
   4:                  1.11.  1,2,4
   5:                 1...1.  1,5
   6:                1..111.  1,2,3,6
   7:               1.....1.  1,7
   8:              1...1.11.  1,2,4,8
   9:             1.....1.1.  1,3,9
  10:            1....1..11.  1,2,5,10
  11:           1.........1.  1,11
  12:          1.....1.1111.  1,2,3,4,6,12
  13:         1...........1.  1,13
  14:        1......1....11.  1,2,7,14
  15:       1.........1.1.1.  1,3,5,15
  16:      1.......1...1.11.  1,2,4,8,16
  17:     1...............1.  1,17
  18:    1........1..1..111.  1,2,3,6,9,18
  19:   1.................1.  1,19
  20:  1.........1....11.11.  1,2,4,5,10,20 (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..999

Crossrefs

Cf. A027750, A055895, A113704.

Programs

Mathematica

Table[If[n == 0, 1, Total[10^Divisors[n]]], {n, 0, 20}] (* or *)
Table[If[n == 0, 1, Sum[If[Mod[n, k] == 0, 10^k, 0], {k, n}]], {n, 0,
  20}] (* or *)
Table[Boole[n == 0] + Total@ MapIndexed[Boole[Divisible[n, #1]]*10^First@ #2 &, Range@ n], {n, 0, 20}] (* or *)
Table[If[n == 0, 1, Function[d, FromDigits @Reverse@ ReplacePart[#, Map[# + 1 -> 1 &, d]] &@ ConstantArray[0, n + 1]]@ Divisors@ n], {n, 0, 20}]
(* Michael De Vlieger, May 24 2017 *)

PARI

a(n) = if (n==0, 1, sum(k=1, n, if (! (n % k), 10^k))); \\ Michel Marcus, May 23 2017

Formula

a(n) = Sum_{k=0..n} if(mod(n, k)=0, 10^k, 0).

G.f.: Sum_{k>=1} 10^k*x^k/(1 - x^k). - Ilya Gutkovskiy, May 23 2017