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

Author

Alon Ran, Jul 19 2020

Keywords

• nonn
• base

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.

Examples

			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).
		

Crossrefs

Cf. A003001, A070565.

Programs

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 *)

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

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

Extensions

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

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