A144376 -id:A144376 - cp's OEIS frontend

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

Sergio Pimentel, Sep 16 2008

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

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

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

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).

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

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 */

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

Edited by Klaus Brockhaus, Sep 19 2008

A385486 Where records occur in A385485.

Original entry on oeis.org

1, 2, 12, 126, 252, 504, 2040, 4080, 16380, 32760, 65520, 524286, 1048572, 4194300, 8388600, 134217720, 268435440, 7516192740, 10737418230, 21474836460, 137438953440, 274877906880, 274877906940, 549755813880, 8796093022200, 8796093022206, 17592186044412

Offset: 1

Amiram Eldar, Jun 30 2025

Chai Wah Wu, Table of n, a(n) for n = 1..27

Cf. A049445, A144376 (decimal analog), A385483, A385485, A385487 (record values).

f[n_] := Module[{m = n, k = 1}, While[Divisible[m, DigitSum[m, 2]], m += 2*n; k += 2]; k];
seq[lim_] := Module[{s = {}, fm = -1, fi}, Do[fi = f[i]; If[fi > fm, fm = fi; AppendTo[s, i]], {i, 1, lim}]; s]; seq[10^4]

f(n) = {my(m = n, k = 1); while(!(m % hammingweight(m)), m += 2*n; k += 2); k;}
list(lim) = my(fm = -1, fi); for(i = 1, lim, fi = f(i); if(fi > fm, fm = fi; print1(i, ", ")));

A385485(a(n)) = A385487(n).

a(21)-a(27) from Chai Wah Wu, Jul 02 2025

A144375 Records in A144262.

Original entry on oeis.org

11, 13, 161, 537, 3611, 537037, 740737, 1005291, 5290873, 216005291, 352733597, 9219175481, 16835016827, 1166734500037

Offset: 1

Klaus Brockhaus, Sep 19 2008

14*10^11 < a(15) <= 6007881003256. [From Donovan Johnson, Jul 22 2010]

Cf. A005349 (Niven numbers), A144262 (smallest k such that k*n is not a Niven number), A144376 (where records occur in A144262).

a(11), a(12) from Klaus Brockhaus, Sep 30 2008

a(13)-a(14) from Donovan Johnson, Jul 22 2010

cp's OEIS Frontend

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

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A385486 Where records occur in A385485.

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A144375 Records in A144262.

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