A336383 - OEIS

%I #26 Sep 14 2020 11:50:40

%S 0,1,21,31,42,52,73,81,319,391,463,583,2911,3667,6451,8793,9927,

%T 237126,254158,278393,2561363,9398143,9431623,9951265,83543869,

%U 83896381,83935261,2843233127,2847297383,2853748583,2885762663,266998137657,685718563667,688373877587

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

%e a(9) = 391 because:

%e    1: 391 - 3*9*1 = 364

%e    2: 364 - 3*6*4 = 292

%e    3: 292 - 2*9*2 = 256

%e    4: 256 - 2*5*6 = 196

%e    5: 196 - 1*9*6 = 142

%e    6: 142 - 1*4*2 = 134

%e    7: 134 - 1*3*4 = 122

%e    8: 122 - 1*2*2 = 118

%e    9: 118 - 1*1*8 = 110

%e After iteration 9, the function becomes idempotent:

%e   10: 110 - 1*1*0 = 110

%e   11: 110 - 1*1*0 = 110

%e   12: 110 - 1*1*0 = 110

%e   ...

%e Additionally, 391 is the smallest number with this property. Thus, it is a(9).

%t nmax = 20; tab = ConstantArray[Null, nmax];

%t For[k = 0, k <= 1000000, k++,

%t  l=Length@  NestWhileList[#-Times @@ IntegerDigits[#] &,k,UnsameQ[##] &, 2]-2;

%t If[tab[[l+1]] == Null, tab[[l+1]] = k]]; tab  (* _Robert Price_, Sep 13 2020 *)

%o (Python)

%o def f(x):

%o     prod = 1

%o     for digit in str(x):

%o         prod *= int(digit)

%o     return x - prod

%o def a(n):

%o     i = 0

%o     iteration = 0

%o     while iteration != n:

%o         i += 1

%o         j = i

%o         iteration = 0

%o         new_j = f(j)

%o         while j != new_j:

%o             iteration += 1

%o             j = new_j

%o             new_j = f(j)

%o     return i

%o (PARI) f(m) = m - vecprod(digits(m)) + (m==0);

%o 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 Cf. A003001, A070565.

%K nonn,base

%O 0,3

%A _Alon Ran_, Jul 19 2020

%E a(27)-a(30) from _Jinyuan Wang_, Aug 14 2020

%E a(31)-a(33) added by _Michael S. Branicky_, Aug 29 2020