A380873 -id:A380873 - cp's OEIS frontend

A380872 Infinite square array, where row r >= 0 is the orbit of r under the map A380873: concatenate(sum of digits, product of digits).

0, 0, 1, 0, 11, 2, 0, 21, 22, 3, 0, 32, 44, 33, 4, 0, 56, 816, 69, 44, 5, 0, 1130, 1548, 1554, 816, 55, 6, 0, 50, 18160, 15100, 1548, 1025, 66, 7, 0, 50, 160, 70, 18160, 80, 1236, 77, 8, 0, 50, 70, 70, 160, 80, 1236, 1449, 88, 9, 0, 50, 70, 70, 70, 80, 1236, 18144, 1664, 99, 10, 0, 50, 70, 70, 70, 80, 1236, 18128, 17144, 1881, 10, 11, 0, 50, 70, 70, 70, 80, 1236, 20128, 17112, 1864

Offset: 0

M. F. Hasler, Apr 01 2025

As usual and required by the "table" display function, the array is read by falling antidiagonals.

			The array starts as follows: (Elements in column 0 are also equal to the row index.)
col.0|  1 |  2 |  3  |  4  |  5  |  6 |  7 |  8  |  9  |  10 |  11 | 12 |  13 |  14
-----+----+----+-----+-----+-----+----+----+-----+-----+-----+-----+----+-----+-----
   0    0     0     0     0     0    0    0     0     0     0     0    0     0     0
   1   11    21    32    56  1130   50   50    50    50    50    50   50    50    50
   2   22    44   816  1548 18160  160   70    70    70    70    70   70    70    70
   3   33    69  1554 15100    70   70   70    70    70    70    70   70    70    70
   4   44   816  1548 18160   160   70   70    70    70    70    70   70    70    70
   5   55  1025    80    80    80   80   80    80    80    80    80   80    80    80
   6   66  1236  1236  1236  1236 1236 1236  1236  1236  1236  1236 1236  1236  1236
   7   77  1449 18144 18128 20128  130   40    40    40    40    40   40    40    40
   8   88  1664 17144 17112  1214   88 1664 17144 17112  1214    88 1664 17144 17112
   9   99  1881  1864 19192 22162 1348 1696 22324  1396 19162 19108  190   100    10
  10   10    10    10    10    10   10   10    10    10    10    10   10    10    10
  11   21    32    56  1130    50   50   50    50    50    50    50   50    50    50
  ...  ...  ...
For example, row 1 is the trajectory of 1 under the map A380873: 1 -> concat (1,1) = 11 -> concat(1+1, 1*1) = 21 -> concat(2+1,2*1) = 32 -> concat(3+2,3*2) = 56 -> ...
Most of the  initial rows reach a fixed point after not too many iterations, but for example row 8 (A271268) and also 38, 83, 88, 146,... reach a cycle of length 5, C(88) = (88, 1664, 17144, 17112, 1214). Another 5-cycle is C(18168) = (18168, 24384, 21768, 24672, 21672), first reached in row 188 and 233.
Fixed points (see A062237) are the multiples of 10 less than 100, and 119 and 1236 (for row 6, 66, 123, ...), 19144 (row 289), and others.

Cf. A380873 (iterated function), A007953 (sum of digits), A007954 (product of digits).

Cf. A271220 (row 6), A271268 (row 8).

A380872_row(r, num_columns=30)=vector(num_columns, i, r=if(i>1, eval(Str(vecsum(r=digits(r)), if(r, vecprod(r)))), r))
A380872_array(rows=9, cols=rows)=Mat(vectorv(rows,i,A380872_row(i-1, cols)))

A(r,0) = r; A(r,n+1) = A380873(A(r,n)) = concat(A007953(A(r,n)), A007954(A(r,n))).

A380871 Limit of the trajectory of n under A380873: concatenate sum and product of digits, if it ends on a fixed point, otherwise the least element of the limit cycle.

Original entry on oeis.org

0, 50, 70, 70, 70, 80, 1236, 40, 88, 10, 10, 50, 50, 60, 20, 50, 70, 50, 70, 10, 20, 50, 70, 50, 70, 80, 70, 10, 80, 90, 30, 60, 50, 70, 60, 90, 90, 40, 88, 90, 40, 20, 70, 60, 70, 20, 70, 40, 70, 10, 50, 50, 80, 90, 20, 80, 50, 50, 80, 40, 60, 70, 70, 90, 70, 50, 1236, 70, 70, 70, 70, 50, 10

Offset: 0

M. F. Hasler, Apr 02 2025

nonn

The fixed points of A380873 are listed in A062237, except for 0.
The first two limit cycles that occur are both of length 5:
* C(88) = (88, 1664, 17144, 17112, 1214), reached for n = 8, 38, 83, 88, ... and
* C(18168) = (18168, 24384, 21768, 24672, 21672), reached for n = 188, 233, ...

			The trajectory of n = 1 under A380873 is: 1 -> concat(1, 1) = 11 -> concat(1+1, 1*1) = 21 -> concat(2+1, 2*1) = 32 -> concat(3+2, 3*2) = 56 -> concat(3+2, 3*2) = 1130 -> concat(1+1+3+0, 1*1*3*0) = 50 -> concat(5+0, 5*0) = 50, so a fixed point is reached, and a(1) = 50.
The trajectory of n = 8 under A380873 is: 8 -> concat(8, 8) = 88 -> concat(8+8, 8*8) = 1664 -> concat(1+6+6+4, 1*6*6*4) = 17144 -> concat(1+7+1+4+4, 1*7*1*4*4) = 17112 -> concat(1+7+1+1+2, 1*7*1*1*2) = 1214 -> concat(1+2+1+4, 1*2*1*4) = 88 -> 1664 etc.: here the limit 5-cycle C(88) = (88, 1664, 17144, 17112, 1214) is reached, so a(8) = min(C(88)) = 88.

Cf. A380873 (iterated function), A007953 (sum of digits), A007954 (product of digits), A062237 (nonzero fixed points of A380873), A380872 (trajectories under A380873).

apply( {A380871(n)=for(i=0,1,my(S=[n]); while(!setsearch(S, n=A380873(n)), S=setunion(S,[n])); i&& n=S[1]);n}, [0..90])

A062237 Numbers k which are (sum of digits of k) concatenated with (product of digits of k).

Original entry on oeis.org

0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 119, 1236, 19135, 19144, 261296, 3634992, 43139968

Offset: 1

Erich Friedman, Jun 30 2001

For a d-digit number with d >= 88, the sum and product of the digits together have fewer than d digits. So every element of this sequence has 87 or fewer digits, hence it is finite. - David W. Wilson, Apr 28 2005

Fixed points of the map A380873: concatenate sum and product of digits. - M. F. Hasler, Apr 01 2025

			1236 has sum of digits 12 and product of digits 36.

Cf. A007953 (sum of digs), A007954 (product of digs), A038364, A038369, A066282, A380873, A380872 (trajectories under map).

sdpdQ[n_]:=Module[{idn=IntegerDigits[n],s,p},s=Total[idn];p=Times@@idn;n==FromDigits[Join[IntegerDigits[s],IntegerDigits[p]]]]; Select[Range[44*10^6],sdpdQ] (* Harvey P. Dale, Nov 23 2024 *)

from math import prod
from sympy.utilities.iterables import multiset_permutations as mp
from itertools import count, islice, combinations_with_replacement as mc
def c(s):
    d = list(map(int, s))
    return sorted(s) == sorted(str(sum(d)) + str(prod(d)))
def ok(s):
    d = list(map(int, s))
    return s[0] != '0' and "".join(s) == str(sum(d)) + str(prod(d))
def nd(d): yield from ("".join(m) for m in mc("0123456789", d))
def b(): yield from (s for d in count(1) for s in nd(d) if c(s))
def a(): yield from (int("".join(p)) for s in b() for p in mp(s) if ok(p))
print(list(islice(a(), 16))) # Michael S. Branicky, Jun 30 2022

More terms from Harvey P. Dale, Jul 04 2001
More terms from David W. Wilson, Apr 28 2005; he reports on May 03 2005 that there are no further terms.
Offset corrected by Altug Alkan, Apr 10 2018

A271220 Concatenate sum of digits of previous term and product of digits of previous term, starting with 6.

Original entry on oeis.org

6, 66, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236, 1236

Offset: 0

Sander Claassen, Apr 02 2016

Each term is created by calculating the sum of the digits of the previous number, and the product of its digits. The results are concatenated to give the new number. Starting with 6, the second number is 66. The third number is generated as follows: 6+6 = 12, 6*6 = 36, which gives 1236. After that, the numbers remain unchanged, because 1+2+3+6 = 12 and 1x2x3x6 = 36, so combined 1236 again.

For more information, see A380873 (the iterated function), A380872 (all trajectories), A062237 (fixed points). - M. F. Hasler, Apr 02 2025

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A380873 (the iterated function), A007953 (sum of digits), A007954 (product of digits), A380872 (all trajectories), A062237 (fixed points).

NestList[FromDigits[Flatten@ {IntegerDigits@ Total@ #, IntegerDigits@ If[Length@ # == 1, #, Times @@ #]}] &@ IntegerDigits@ # &, 6, 50] (* Michael De Vlieger, Apr 02 2016 *)

A380872_row(6) \\ M. F. Hasler, Apr 02 2025

a(n) = 1236 for all n > 2. - M. F. Hasler, Apr 02 2025

Offset changed to 0 by M. F. Hasler, Apr 02 2025

cp's OEIS Frontend

A380872 Infinite square array, where row r >= 0 is the orbit of r under the map A380873: concatenate(sum of digits, product of digits).

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A380871 Limit of the trajectory of n under A380873: concatenate sum and product of digits, if it ends on a fixed point, otherwise the least element of the limit cycle.

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A062237 Numbers k which are (sum of digits of k) concatenated with (product of digits of k).

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A271220 Concatenate sum of digits of previous term and product of digits of previous term, starting with 6.

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