A336383 a(n) is the smallest number such that, with f(x) = x - (the product of the digits of x), f(a(n)) reaches a fixed point after n iterations.

0, 1, 21, 31, 42, 52, 73, 81, 319, 391, 463, 583, 2911, 3667, 6451, 8793, 9927, 237126, 254158, 278393, 2561363, 9398143, 9431623, 9951265, 83543869, 83896381, 83935261, 2843233127, 2847297383, 2853748583, 2885762663, 266998137657, 685718563667, 688373877587

Offset

0,3

Comments

A fixed point occurs once the function returns a number that contains the digit 0. After that, the product of the digits will be 0, and so subtracting it from the number will be idempotent.

This sequence is conceptually similar to A003001, though unlike the latter, it is probably infinite.

Links

Table of n, a(n) for n=0..33.

Example

a(9) = 391 because:
   1: 391 - 3*9*1 = 364
   2: 364 - 3*6*4 = 292
   3: 292 - 2*9*2 = 256
   4: 256 - 2*5*6 = 196
   5: 196 - 1*9*6 = 142
   6: 142 - 1*4*2 = 134
   7: 134 - 1*3*4 = 122
   8: 122 - 1*2*2 = 118
   9: 118 - 1*1*8 = 110
After iteration 9, the function becomes idempotent:
  10: 110 - 1*1*0 = 110
  11: 110 - 1*1*0 = 110
  12: 110 - 1*1*0 = 110
  ...
Additionally, 391 is the smallest number with this property. Thus, it is a(9).

Mathematica

nmax = 20; tab = ConstantArray[Null, nmax];
For[k = 0, k <= 1000000, k++,
 l=Length@  NestWhileList[#-Times @@ IntegerDigits[#] &, k, UnsameQ[##] &, 2]-2;
If[tab[[l+1]] == Null, tab[[l+1]] = k]]; tab  (* Robert Price, Sep 13 2020 *)

Prog

(Python)
def f(x):
    prod = 1
    for digit in str(x):
        prod *= int(digit)
    return x - prod
def a(n):
    i = 0
    iteration = 0
    while iteration != n:
        i += 1
        j = i
        iteration = 0
        new_j = f(j)
        while j != new_j:
            iteration += 1
            j = new_j
            new_j = f(j)
    return i
(PARI) f(m) = m - vecprod(digits(m)) + (m==0);
lista(nn) = {my(c, m, t); for(k=0, nn, c=0; m=k; while(m!=(m=f(m)), c++); if(c==t, print1(k, ", "); t++)); } \\ Jinyuan Wang, Aug 14 2020

Crossrefs

Cf. A003001, A070565.

Sequence in context: A342838 A261322 A123846 * A168000 A031889 A034101

Adjacent sequences: A336380 A336381 A336382 * A336384 A336385 A336386

Keyword

nonn,base

Author

Alon Ran, Jul 19 2020

Extensions

a(27)-a(30) from Jinyuan Wang, Aug 14 2020

a(31)-a(33) added by Michael S. Branicky, Aug 29 2020

Status

approved