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%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