A352871 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.

-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, -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, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0

Offset

1

Comments

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

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.

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

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

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

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

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

References

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

Links

Table of n, a(n) for n=1..78.

Example

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.

Maple

a:= proc(n) option remember;
      `if`(irem(n, add(i, i=convert(n, base, 10)), 'm')>0, 0,
      `if`(n=m, -1, (t-> `if`(t<0, t, t+1))(a(m))))
    end:
seq(a(n), n=1..102);  # Alois P. Heinz, Apr 08 2022

Prog

(Python)
def sumdigits(n: int) -> int:
    return sum(map(int, str(n)))
def a(n: int) -> int:
    i = 0
    while True:
        denom = sumdigits(n)
        if denom == 1:
            i = -1
            break
        elif n % denom == 0:
            i = i + 1
            n = n // denom
        else:
            break
    return i

Crossrefs

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

Sequence in context: A105586 A202022 A136522 * A188641 A086299 A131364

Adjacent sequences: A352868 A352869 A352870 * A352872 A352873 A352874

Keyword

sign,base,easy

Author

Jonathan Berliner, Apr 06 2022

Status

approved