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A144262
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a(n) = smallest k such that k*n is not a Niven (or Harshad) number.
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8
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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
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OFFSET
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1,1
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COMMENTS
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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.
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 O6 2014
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LINKS
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EXAMPLE
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a(2) = 7 since 2, 4, 6, 8, 10 and 12 are all Niven numbers; but 7*2 = 14 is not.
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MATHEMATICA
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a[n_] := Module[{k = 1}, While[Divisible[k*n, Plus @@ IntegerDigits[k*n]], k++]; k]; Array[a, 100] (* Amiram Eldar, Sep 05 2020 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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base,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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