A144262 a(n) = smallest k such that k*n is not a Niven (or Harshad) number.

11, 7, 5, 4, 3, 11, 2, 2, 11, 13, 1, 8, 1, 1, 1, 1, 1, 161, 1, 8, 5, 1, 1, 4, 1, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 83, 1, 1, 1, 4, 1, 4, 1, 1, 11, 1, 1, 2, 1, 5, 1, 1, 1, 537, 1, 1, 1, 1, 1, 83, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 68, 1, 1, 1, 1, 1, 1, 1, 2, 7, 1, 1, 2, 1, 1, 1, 1, 1, 211, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Sergio Pimentel, Sep 16 2008

Keywords

• base
• nonn

Comments

Niven (or Harshad) numbers are numbers that can be divided by the sum of their digits.

If n is not a Niven number then a(n) is obviously 1. Some terms are rather large: a(108) = 3611, a(540) = 537037; see also A144375 and A144376.

Does a(n) exist for all n? - Klaus Brockhaus, Sep 19 2008

a(n) should exist for all n since the density of the Niven numbers is zero and it has been proved that arbitrarily large gaps exist between Niven numbers. [Sergio Pimentel, Sep 20 2008]

Let N be the number formed by concatenating R copies of n, where R is the smallest power of 10 that exceeds n. Then N is a multiple of n, but not a Niven number; since R divides the sum of the digits of N, but R does not divide N. - David Radcliffe, Oct 06 2014

Examples

			a(2) = 7 since 2, 4, 6, 8, 10 and 12 are all Niven numbers; but 7*2 = 14 is not.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Harshad Number

Crossrefs

Cf. A005349 (Niven numbers), A144261 (smallest k such that k*n is a Niven number), A144375 (records in A144262), A144376 (where records occur in A144262).

Programs

Mathematica

a[n_] := Module[{k = 1}, While[Divisible[k*n, Plus @@ IntegerDigits[k*n]], k++]; k]; Array[a, 100] (* Amiram Eldar, Sep 05 2020 *)

PARI

digitsum(n) = {local(s=0); while(n, s+=n%10; n\=10); s}
{for(n=1, 100, k=1; while((p=k*n)%digitsum(p)==0, k++); print1(k, ","))} /* Klaus Brockhaus, Sep 19 2008 */

Python

def a(n):
    kn = n
    while kn % sum(map(int, str(kn))) == 0: kn += n
    return kn//n
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Nov 07 2021

Extensions

Edited by Klaus Brockhaus, Sep 19 2008