A316650 Result when n is divided by the sum of its digits and the resulting integer is concatenated to the remainder.

10, 10, 10, 10, 10, 10, 10, 10, 10, 100, 51, 40, 31, 24, 23, 22, 21, 20, 19, 100, 70, 52, 43, 40, 34, 32, 30, 28, 27, 100, 73, 62, 53, 46, 43, 40, 37, 35, 33, 100, 81, 70, 61, 54, 50, 46, 43, 40, 310, 100, 83, 73, 65, 60, 55, 51, 49, 46, 43, 100, 85, 76, 70, 64, 510, 56, 52, 412, 49, 100, 87

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, Jul 09 2018

Keywords

• base
• nonn
• look

Comments

When the remainder is zero, this 0 is still concatenated to the result (12/3 becomes 40).

All terms of A052224 are fixed points; more generally, if n has digital sum 10^k for some k > 0 and the remainder of n when divided by 10^k has k decimal digits, then n is a fixed point (corrected by Rémy Sigrist, Jul 10 2018).

Examples

			1 divided by 1 is 1 with the remainder 0, thus a(1) = 10;
2 divided by 2 is 1 with the remainder 0, thus a(2) = 10;
3 divided by 3 is 1 with the remainder 0, thus a(3) = 10;
4 divided by 4 is 1 with the remainder 0, thus a(4) = 10;
...
10 divided by (1+0) is 10 with the remainder 0, thus a(10) = 100;
11 divided by (1+1) is 5 with the remainder 1, thus a(11) = 51;
12 divided by (1+2) is 4 with the remainder 0, thus a(12) = 40;
13 divided by (1+3) is 3 with the remainder 1, thus a(13) = 31;
...
2018 divided by (2+0+1+8) is 183 with the remainder 5, thus a(2018) = 1835.
Etc.
		

Links

• Jean-Marc Falcoz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A007953, A052224.

Programs

Mathematica

Array[FromDigits@ Flatten[IntegerDigits@ # & /@ QuotientRemainder[#, Total[IntegerDigits@ #]]] &, 71] (* Michael De Vlieger, Jul 10 2018 *)

PARI

a(n, base=10) = my (ds=sumdigits(n, base), q=n\ds, r=n%ds); q * base^max(1, #digits(r, base)) + r \\ Rémy Sigrist, Jul 10 2018

Formula

a(10^k) = 10^(k+1) for any k >= 0. - Rémy Sigrist, Jul 10 2018