A389235 Number of repeating Harshad algorithm steps until a non-Harshad number or a power of 10 is reached.

0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset

1,12

Comments

a(n) is the number of Harshad algorithm steps that can be done starting with n and always reaching another Harshad number (See also A114440). Stopping points are non-Harshad numbers or Harshad numbers that are a powers of 10 (See example).

Links

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Brady Haran and Tony Padilla, The Joy of Harshad Numbers, YouTube video, 2025.

Eric Weisstein, Harshad Numbers.

Wikipedia, Harshad number.

Example

a(20) = 1 because digit sum of 20 is 2 and 20/2 = 10.
10 is in fact a Harshad number too, but a Harshad number with an infinite cycle.
So we have to stop here. This happens for all powers of 10.
So a(1), a(10), a(100) ... are always 0.

Mathematica

a[n_] := -1 + Length[NestWhileList[# / DigitSum[#] &, n, DigitSum[#] > 1 && Divisible[#, DigitSum[#]] &]]; Array[a, 100] (* Amiram Eldar, Sep 27 2025 *)

Prog

(Python)
def A389235(n):
    c = 0
    while not n % (digsum:=sum([int(d) for d in str(n)])) and digsum != 1: n, c = n // digsum, c + 1
    return c
print([A389235(n) for n in range(1, 100)])
(PARI) a(n) = {my(k=0, r=n, s); while((s=sumdigits(r)) != 1 && r%s==0, r/=s; k++); k} \\ Andrew Howroyd, Sep 26 2025

Crossrefs

Cf. A007953, A005349, A114440, A188641, A235507, A235697, A236338, A385105, A389220.

Sequence in context: A341620 A097798 A065205 * A036272 A257403 A326559

Adjacent sequences: A389232 A389233 A389234 * A389236 A389237 A389238

Keyword

nonn,base,easy,changed

Author

Karl-Heinz Hofmann, Sep 26 2025

Status

approved