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A144261 a(n) = smallest k such that k*n is a Niven (or Harshad) number.

%I A144261 #23 Feb 16 2025 08:33:09
%S A144261 1,1,1,1,1,1,1,1,1,1,10,1,9,3,2,3,6,1,6,1,1,5,9,1,2,6,1,3,9,1,12,6,4,
%T A144261 3,2,1,3,3,3,1,10,1,12,3,1,5,9,1,8,1,2,3,18,1,2,2,2,9,9,1,12,6,1,3,3,
%U A144261 2,3,3,3,1,18,1,7,3,2,2,4,2,9,1,1,5,18,1,6,6,3,3,9,1,4,5,4,9,2,2,12,4,2,1
%N A144261 a(n) = smallest k such that k*n is a Niven (or Harshad) number.
%C A144261 Niven (or Harshad) numbers are numbers that are divisible by the sum of their digits.
%C A144261 Does a(n) exist for all n? - _Klaus Brockhaus_, Sep 19 2008
%H A144261 David Radcliffe, <a href="/A144261/b144261.txt">Table of n, a(n) for n = 1..10000</a>
%H A144261 David Radcliffe, <a href="/A144261/a144261.pdf">Every positive integer divides a Harshad number</a>
%H A144261 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarshadNumber.html">Harshad Number</a>
%e A144261 a(14) = 3 since neither 1*14 or 2*14 are Niven numbers, but 3*14 = 42 is a Niven number: 42 = 7*(4+2).
%t A144261 niv[n_]:=Module[{k=1},While[!Divisible[k*n,Total[IntegerDigits[ k*n]]], k++]; k]; Array[niv,100] (* _Harvey P. Dale_, Jul 23 2016 *)
%o A144261 (PARI) digitsum(n) = {local(s=0); while(n, s+=n%10; n\=10); s}
%o A144261 {for(n=1, 100, k=1; while((p=k*n)%digitsum(p)>0, k++); print1(k, ","))} /* _Klaus Brockhaus_, Sep 19 2008 */
%o A144261 (Python)
%o A144261 from itertools import count
%o A144261 def A144261(n): return next(filter(lambda k:not (m:=k*n) % sum(int(d) for d in str(m)), count(1))) # _Chai Wah Wu_, Nov 04 2022
%Y A144261 Cf. A005349 (Niven numbers), A144262 (smallest k such that k*n is not a Niven number), A144363 (records in A144261), A144364 (where records occur in A144261).
%K A144261 base,nonn
%O A144261 1,11
%A A144261 _Sergio Pimentel_, Sep 16 2008
%E A144261 Edited and extended by _Klaus Brockhaus_, Sep 19 2008