A380873 Concatenate sum and product of decimal digits of n.

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 20, 32, 44, 56, 68, 710, 812, 914, 1016, 1118, 30, 43, 56, 69, 712, 815, 918, 1021, 1124, 1227, 40, 54, 68, 712, 816, 920, 1024, 1128, 1232, 1336, 50, 65, 710, 815, 920, 1025, 1130, 1235, 1340, 1445, 60

Offset: 0

Author

M. F. Hasler, Apr 01 2025

Keywords

• nonn
• base

Comments

This sequence is motivated by A271220 and A271268 (maybe others?) which give the trajectory of specific starting values under iterations of this map.

The fixed points of this map, (0, 10, ..., 90, 119, 1236, ...), are listed in sequence A062237 (except for 0).

Besides the fixed points, this map has also limiting cycles, for example:

* the cycle C(88) = (88, 1664, 17144, 17112, 1214) of length 5, first reached for initial values 8 (cf. A271268) and 38, 83, 88, 146, ....

* Another 5-cycle is C(18168) = (18168, 24384, 21768, 24672, 21672), first reached for initial values 188 and 233.

The infinite square array A380872 gives the trajectory of starting value r = 0, 1, 2, ... in row r.

Examples

			For n = 0, ..., 9, a(n) = 11*n because sum and product of digits of n are equal to n.
a(10) = concat(1+0, 1*0) = 10, a(11) = concat(1+1, 1*1) = 21, a(12) = concat(1+2, 1*2) = 32, etc.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A007953 (sum of digits), A007954 (product of digits).

Cf. A062237 (fixed points of this map).

Cf. A271220 (trajectory of 6), A271268 (trajectory of 8), A380872 (trajectories of all nonnegative integers as rows of a table).

Programs

Maple

a:= n-> (l-> parse(cat(add(i, i=l), mul(i, i=l))))(convert(n, base, 10)):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 12 2025

Mathematica

a[n_]:=FromDigits[Join[IntegerDigits[Total[IntegerDigits[n]]],IntegerDigits[Times@@IntegerDigits[n]]]];Array[a,61,0] (* James C. McMahon, Apr 02 2025 *)

PARI

apply( {A380873(n)=eval(Str(vecsum(n=digits(n)),if(n,vecprod(n))))}, [0..99])

Python

from math import prod
def A380873(n): return int(f"{sum(d:=list(map(int,str(n))))}{prod(d)}")
print(first_99 := list(map(A380873,range(99))))