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

0,2

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.

Links

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

Example

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.

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 *)

Prog

(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))))

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).

Sequence in context: A351877 A073729 A073730 * A367610 A087346 A060314

Adjacent sequences: A380870 A380871 A380872 * A380874 A380875 A380876

Keyword

nonn,base

Author

M. F. Hasler, Apr 01 2025

Status

approved