A352871 - OEIS

%I #57 Aug 04 2022 15:45:58

%S -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,-1,0,0,0,0,0,-1,0,-1,-1,0,0,-1,0,0,

%T -1,0,0,-1,0,0,0,0,0,-1,0,0,0,-1,0,-1,0,0,-1,0,0,-1,0,-1,0,0,0,-1,0,0,

%U 0,0,0,-1,0,0,-1,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0

%N a(n) is the number of iterations, starting with x = n, which can be made of x -> x/sumdigits(x) with x remaining an integer, or -1 if x remains an integer through infinitely many iterations.

%C sumdigits(x) = A007953(x) is the sum of the decimal digits of x.

%C A Harshad number (A005349) is divisible by the sum of its digits and the iterations here step through Harshad numbers until reaching a non-Harshad number.

%C a(102) = 1 is the first term that is neither 0 nor -1.

%C a(1008) is the first term whose value is 2.

%C a(10080) is the first term whose value is 3.

%C a(100800) is the first term whose value is 4.

%C a(1008000) is the first term whose value is 5.

%D D. R. Kaprekar, Multidigital Numbers, Scripta Mathematica 21 (1955), 27.

%e 13000/(1+3+0+0+0) = 3250, and 3250/(3+2+5+0) = 325, but 325/(3+2+5) is not an integer, so a(13000) = 2.

%p a:= proc(n) option remember;

%p       `if`(irem(n, add(i, i=convert(n, base, 10)), 'm')>0, 0,

%p       `if`(n=m, -1, (t-> `if`(t<0, t, t+1))(a(m))))

%p     end:

%p seq(a(n), n=1..102);  # _Alois P. Heinz_, Apr 08 2022

%o (Python)

%o def sumdigits(n: int) -> int:

%o     return sum(map(int, str(n)))

%o def a(n: int) -> int:

%o     i = 0

%o     while True:

%o         denom = sumdigits(n)

%o         if denom == 1:

%o             i = -1

%o             break

%o         elif n % denom == 0:

%o             i = i + 1

%o             n = n // denom

%o         else:

%o             break

%o     return i

%Y Cf. A005349, A007953, A065877, A188641, A235507, A235697.

%K sign,base,easy

%O 1

%A _Jonathan Berliner_, Apr 06 2022