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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.

%I A336383 #26 Sep 14 2020 11:50:40
%S A336383 0,1,21,31,42,52,73,81,319,391,463,583,2911,3667,6451,8793,9927,
%T A336383 237126,254158,278393,2561363,9398143,9431623,9951265,83543869,
%U A336383 83896381,83935261,2843233127,2847297383,2853748583,2885762663,266998137657,685718563667,688373877587
%N 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.
%C A336383 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.
%C A336383 This sequence is conceptually similar to A003001, though unlike the latter, it is probably infinite.
%e A336383 a(9) = 391 because:
%e A336383    1: 391 - 3*9*1 = 364
%e A336383    2: 364 - 3*6*4 = 292
%e A336383    3: 292 - 2*9*2 = 256
%e A336383    4: 256 - 2*5*6 = 196
%e A336383    5: 196 - 1*9*6 = 142
%e A336383    6: 142 - 1*4*2 = 134
%e A336383    7: 134 - 1*3*4 = 122
%e A336383    8: 122 - 1*2*2 = 118
%e A336383    9: 118 - 1*1*8 = 110
%e A336383 After iteration 9, the function becomes idempotent:
%e A336383   10: 110 - 1*1*0 = 110
%e A336383   11: 110 - 1*1*0 = 110
%e A336383   12: 110 - 1*1*0 = 110
%e A336383   ...
%e A336383 Additionally, 391 is the smallest number with this property. Thus, it is a(9).
%t A336383 nmax = 20; tab = ConstantArray[Null, nmax];
%t A336383 For[k = 0, k <= 1000000, k++,
%t A336383  l=Length@  NestWhileList[#-Times @@ IntegerDigits[#] &,k,UnsameQ[##] &, 2]-2;
%t A336383 If[tab[[l+1]] == Null, tab[[l+1]] = k]]; tab  (* _Robert Price_, Sep 13 2020 *)
%o A336383 (Python)
%o A336383 def f(x):
%o A336383     prod = 1
%o A336383     for digit in str(x):
%o A336383         prod *= int(digit)
%o A336383     return x - prod
%o A336383 def a(n):
%o A336383     i = 0
%o A336383     iteration = 0
%o A336383     while iteration != n:
%o A336383         i += 1
%o A336383         j = i
%o A336383         iteration = 0
%o A336383         new_j = f(j)
%o A336383         while j != new_j:
%o A336383             iteration += 1
%o A336383             j = new_j
%o A336383             new_j = f(j)
%o A336383     return i
%o A336383 (PARI) f(m) = m - vecprod(digits(m)) + (m==0);
%o A336383 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
%Y A336383 Cf. A003001, A070565.
%K A336383 nonn,base
%O A336383 0,3
%A A336383 _Alon Ran_, Jul 19 2020
%E A336383 a(27)-a(30) from _Jinyuan Wang_, Aug 14 2020
%E A336383 a(31)-a(33) added by _Michael S. Branicky_, Aug 29 2020