A385105 Harshad root of integer n: the largest integer produced by repeatedly dividing n by its digit sum until it fails to produce another integer or it reaches a power of 10.

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 1, 13, 14, 15, 16, 17, 1, 19, 10, 1, 22, 23, 1, 25, 26, 1, 28, 29, 10, 31, 32, 33, 34, 35, 1, 37, 38, 39, 10, 41, 1, 43, 44, 1, 46, 47, 1, 49, 10, 51, 52, 53, 1, 55, 56, 57, 58, 59, 10, 61, 62, 1, 64, 65, 66, 67, 68, 69, 10, 71, 1, 73, 74, 75, 76, 77, 78, 79

Offset

1,10

Comments

a(102) = 34 is the least integer without itself or a power of 10 as a root.

Links

James Carruthers, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Harshad Numbers.

Wikipedia, Harshad number.

Example

a(20) = 10 since 20/(2+0) = 10 and 10 loops to itself.

Mathematica

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

Prog

(Python)
def a(n):
    while n % (digsum:=sum([int(d) for d in str(n)])) == 0 and digsum != 1:
        n//=digsum
    return n
print([a(n) for n in range(1, 80)])
(PARI) a(n) = {my(r=n, s); while((s=sumdigits(r)) != 1 && r%s==0, r/=s); r} \\ Andrew Howroyd, Sep 26 2025

Crossrefs

Cf. A005349 (Harshad numbers), A188641 (characteristic function of Harshad numbers), A389235 (final values).

Sequence in context: A324660 A336035 A334676 * A176998 A328752 A173821

Adjacent sequences: A385102 A385103 A385104 * A385106 A385107 A385108

Keyword

nonn,base,easy

Author

James Carruthers, Sep 25 2025

Status

approved