A144261 -id:A144261 - cp's OEIS frontend

A144364 Where records occur in A144261.

1, 11, 31, 53, 331, 377, 983, 1499, 2609, 3329, 6637, 6997, 19937, 34987, 157961, 173699, 256661, 1349923, 1616359, 1993333, 2199833, 5794969, 6906869, 12204431, 14223073, 20666551, 168499997, 4819002127, 7331752277, 8259275501, 27017515331, 27581017543

Offset: 1

Klaus Brockhaus, Sep 19 2008

Amiram Eldar, Table of n, a(n) for n = 1..39

Cf. A005349 (Niven numbers), A144261 (smallest k such that k*n is a Niven number), A144363 (records in A144261).

f[n_] := Module[{k = 1, m = n}, While[!Divisible[m, DigitSum[m]], m +=n; k++]; k]; seq[lim_] := Module[{s = {}, fm = 0, fi}, Do[fi = f[i]; If[fi > fm, fm = fi; AppendTo[s, i]], {i, 1, lim}]; s]; seq[10^5] (* Amiram Eldar, Jun 30 2025 *)

f(n) = {my(k = 1, m = n); while(m % sumdigits(m), m +=n; k++); k;}
list(lim) = my(fm = 0, fi); for(i = 1, lim, fi = f(i); if(fi > fm, fm = fi; print1(i, ", "))); \\ Amiram Eldar, Jun 30 2025

a(27)-a(30) from Donovan Johnson, Jul 20 2010

a(31)-a(32) from Amiram Eldar, Jun 30 2025

A144363 Records in A144261.

Original entry on oeis.org

1, 10, 12, 18, 21, 30, 42, 48, 54, 60, 72, 96, 108, 120, 130, 150, 165, 168, 180, 192, 210, 216, 240, 252, 280, 322, 441, 462, 472, 486, 488, 520, 528, 536, 560, 600, 630, 729, 1188

Offset: 1

Klaus Brockhaus, Sep 19 2008

Cf. A005349 (Niven numbers), A144261 (smallest k such that k*n is a Niven number), A144364 (where records occur in A144261).

f[n_] := Module[{k = 1, m = n}, While[!Divisible[m, DigitSum[m]], m +=n; k++]; k]; seq[lim_] := Module[{s = {}, fm = 0, fi}, Do[fi = f[i]; If[fi > fm, fm = fi; AppendTo[s, fi]], {i, 1, lim}]; s]; seq[10^5] (* Amiram Eldar, Jun 30 2025 *)

f(n) = {my(k = 1, m = n); while(m % sumdigits(m), m +=n; k++); k;}
list(lim) = my(fm = 0, fi); for(i = 1, lim, fi = f(i); if(fi > fm, fm = fi; print1(fi, ", "))); \\ Amiram Eldar, Jun 30 2025

a(27)-a(30) from Donovan Johnson, Jul 20 2010

a(31)-a(39) from Amiram Eldar, Jun 30 2025

A358067 a(n) is the smallest m such that A144261(m) = n.

Original entry on oeis.org

1, 15, 14, 33, 22, 17, 73, 49, 13, 11, 529, 31, 397, 293, 241, 199, 1633, 53, 3727, 761, 331, 491, 4343, 431, 1943, 887, 383, 3659, 3809, 377, 15863, 9419, 2713, 2993, 26753, 1583, 30311, 5297, 8971, 2753, 5363, 983, 11603, 4919, 18314, 14657, 59303, 1499, 99179

Offset: 1

Bernard Schott, Oct 29 2022

In other words, a(n) is the smallest element m of the set of the integers k that satisfy {A144261(k) = n and n * k is a Niven number} (see Example section).

Same question as in A144261: does a(n) exist for all n?

			A144261(15) = A144261(25) = A144261(35) = A144261(51) = ... = 2 because 15, 25, 35, 51, ... are not Niven numbers and 2 is the smallest integer such that 2*15=30, 2*25=50, 2*35=70, 2*51=102, ..., are Niven (or Harshad) numbers. Since 15 is the least integer that, when multiplied by 2, yields a Niven number (2*15), a(2) = 15.

Giovanni Resta, Harshad numbers.

Cf. A005349 (Niven numbers), A144261.

f[n_] := Module[{k = 1, p}, While[! Divisible[p = k*n, Plus @@ IntegerDigits[p]], k++]; k]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50, 10^6] (* Amiram Eldar, Oct 29 2022 *)

f(n) = my(k=1); while ((k*n) % sumdigits(k*n), k++); k; \\ A144261
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Oct 31 2022

from itertools import count
def A358067(n): return next(filter(lambda m:n==next(filter(lambda k:not (r:=k*m) % sum(int(d) for d in str(r)), count(1))),count(1))) # Chai Wah Wu, Nov 04 2022

More terms from Amiram Eldar, Oct 29 2022

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

Original entry on oeis.org

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

A226169 Niven numbers when expressed in bases 1 through 10.

Original entry on oeis.org

1, 2, 4, 6, 24, 40, 48, 72, 120, 144, 180, 216, 252, 288, 324, 336, 360, 432, 504, 576, 648, 720, 756, 780, 840, 960, 1008, 1056, 1080, 1092, 1200, 1260, 1296, 1344, 1380, 1440, 1512, 1584, 1620, 1680, 1728, 1764, 1800, 1944, 2016, 2196, 2304, 2352, 2448

Offset: 1

Sergio Pimentel, May 29 2013

The first 10 odd terms greater than 1 are a(1151) = 543375, 5329233, 18640125, 19178775, 23186625, 30131535, 35026425, 36797775, 46101825, 51856875. - Giovanni Resta, Jun 01 2013

			Example: 336 is in the sequence because the sum of digits of 336 when expressed in bases 1 through 10 is: 336, 3, 4, 3, 8, 6, 12, 7, 8, 12; and 336 is divisible by all these numbers.  In this particular example 336 keeps this property in bases 11, 12 and 13, but not 14.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049445, A118363, A005349, A144261, A144262.

Select[Range[10^4], Catch[Do[If[Mod[#, Total@IntegerDigits[#, b]] > 0, Throw@ False], {b, 2, 10}]; True] &] (* Giovanni Resta, May 29 2013 *)
t = Table[b = 2; While[s = Total[IntegerDigits[n, b]]; s < n && Mod[n, s] == 0, b++]; If[s == n, b = 0]; b, {n, 2000}]; Flatten[Position[t, ?(# == 0 || # > 10 &)]] (* _T. D. Noe, May 30 2013 *)

Missing a(17) and a(35)-a(49) from Giovanni Resta, May 29 2013

A385482 a(n) is the least number k such that k*n is a binary Niven number (A049445).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 5, 1, 10, 3, 4, 1, 2, 1, 12, 1, 1, 3, 3, 1, 12, 5, 3, 3, 4, 2, 5, 1, 2, 1, 12, 1, 5, 6, 4, 1, 5, 1, 4, 3, 4, 2, 12, 1, 12, 6, 4, 3, 4, 2, 1, 3, 4, 2, 5, 1, 6, 5, 2, 1, 2, 1, 12, 1, 1, 6, 4, 1, 6, 3, 4, 3, 4, 2, 5, 1, 1, 3, 4, 1, 4

Offset: 1

Amiram Eldar, Jun 30 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049445, A144261 (decimal analog), A363788, A385483 (indices of records), A385484 (record values), A385485.

a[n_] := Module[{m = n, k = 1}, While[!Divisible[m, DigitSum[m, 2]], m += n; k++]; k]; Array[a, 100]

a(n) = {my(m = n, k = 1); while(m % hammingweight(m), m += n; k++); k;}

from itertools import count
def a(n): return next(k for k in count(1) if (m:=k*n)%m.bit_count() == 0)
print([a(n) for n in range(1, 86)]) # Michael S. Branicky, Jun 30 2025

a(n) = 1 if and only if n is in A049445.

a(n) = 2 if and only if 2*n is in A363788.

A226171 Smallest base in which n is not Niven (or zero if n is Niven in every base).

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 2, 6, 2, 4, 2, 8, 2, 2, 2, 6, 2, 8, 2, 7, 5, 2, 2, 14, 2, 2, 2, 2, 2, 2, 2, 6, 2, 3, 2, 8, 2, 2, 2, 12, 2, 3, 2, 2, 2, 2, 2, 14, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 8, 2, 2, 2, 6, 2, 3, 2, 3, 3, 2, 2, 14, 2, 2, 2, 2, 2, 2, 2, 8, 5, 2, 2, 5, 2, 2

Offset: 1

Sergio Pimentel, May 29 2013

Niven numbers (in base b) are divisible by the sum of their digits (in base b).
Questions: are 1, 2, 4 and 6 the only zeros in this sequence? Where are the records or high water marks?
From Bert Dobbelaere, Oct 08 2018: (Start)
1,2,4,6 are the only numbers that are Niven in every base.
Proof: Suppose n is Niven in every base, then consider the base-b representations of n for (n/2) < b <= n. These are all 2-digit numbers with 1 as 1st digit and (n-b) as last digit. Then 1+n-b is a divisor of n for all b, meaning that all numbers between 1 up to n/2 are divisors of n. Clearly there are no such numbers larger than 6.
a(n) < 60 for n < 10^13.
(End)

			The sum of digits of 24 in bases 1 through 14 are:  24, 2, 4, 3, 8, 4, 6, 3, 8, 6, 4, 2, 12, 11.  24 is divisible by all these numbers except the last one; therefore a(24) = 14.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A049445, A118363, A005349, A144261, A144262, A226169.

Cf. A225427 (least Niven number for all bases from 1 to n).

Table[b = 2; While[s = Total[IntegerDigits[n, b]]; s < n && Mod[n, s] == 0, b++]; If[s == n, b = 0]; b, {n, 100}] (* T. D. Noe, May 30 2013 *)

a(n) = {for (b=2, n-1, if (frac(n/sumdigits(n,b)), return(b));); 0;} \\ Michel Marcus, Oct 23 2018

A357936 a(n) is the least multiple of n that is a Niven (or Harshad) number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 110, 12, 117, 42, 30, 48, 102, 18, 114, 20, 21, 110, 207, 24, 50, 156, 27, 84, 261, 30, 372, 192, 132, 102, 70, 36, 111, 114, 117, 40, 410, 42, 516, 132, 45, 230, 423, 48, 392, 50, 102, 156, 954, 54, 110, 112, 114, 522, 531, 60

Offset: 1

Rémy Sigrist, Oct 21 2022

Niven (or Harshad) numbers are divisible by the sum of their digits, and correspond to sequence A005349.

			For n = 67, we have:
.
      k  67*k  Niven?
      -  ----  ------
      1    67  No
      2   134  No
      3   201  Yes
so a(67) = 201.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
David Radcliffe, Every positive integer divides a Harshad number
Index entries for sequences related to decimal expansion of n

Cf. A005349, A144261, A357937.

a[n_]:=Module[{k=1}, While[!Divisible[k*n, Total[IntegerDigits[k*n]]], k++]; k*n]; Array[a, 60]

a(n, base=10) = forstep (m=n, oo, n, if (m%sumdigits(m, base)==0, return (m)))

a(n) = n * A144261(n).

A144378 Initial term of a series of exactly n consecutive non-Niven (or Harshad) numbers.

Original entry on oeis.org

11, 22, 37, 136, 13, 64, 73, 163, 91, 1730, 289, 1639, 379, 1660, 2737, 919, 559, 14878, 7561, 5671, 9753, 2890, 7777, 4888, 5785, 5590, 27973, 47872, 28681, 22681, 3785, 36184, 46281, 71281, 6481, 48952, 48763, 64978, 119773, 69782, 77881, 55973

Offset: 1

Sergio Pimentel, Sep 18 2008

Multiples of 18 seem to be the high water marks, while terms of the form 18n - 1 seem to be the valleys of this sequence.
Many terms end in '81' for some reason.
This sequence is analog to A060159 with non-Niven numbers.
This sequence is infinite, as opposed as A060159.

			a(5) = 13 since 13, 14, 15, 16 and 17 are all non-Niven numbers and this is the first occurrence of exactly 5 non-Niven numbers.

Klaus Brockhaus, Table of n, a(n) for n=1..125
Eric Weisstein's World of Mathematics, Harshad Number

Cf. A005349, A082516, A060159, A144261, A144262.

digitsum(n) = {local(s=0); while(n, s+=n%10; n\=10); s}
{m=120000; z=42; w=vector(z); n=1; while(n<=m, while(n%digitsum(n)==0, n++); a=n; c=0; while(n%digitsum(n)>0, n++; c++); if(c<=z&&w[c]==0, w[c]=a)); j=1; while(j<=z&&w[j]>0, print1(w[j], ","); j++)} /* Klaus Brockhaus, Sep 24 2008 */

a(2), a(22), a(42) corrected by Klaus Brockhaus, Sep 24 2008

A337731 a(n) is the smallest k >= 1 such that k*n is a Moran number.

Original entry on oeis.org

18, 9, 6, 21, 9, 3, 3, 19, 2, 19, 18, 7, 9, 3, 3, 37, 9, 1, 6, 199, 1, 9, 9, 37, 199, 6, 1, 3, 9, 1663, 12, 937, 6, 1117, 1657, 1361, 3, 3, 3, 17497, 18, 1, 12, 10909, 1, 14563, 9, 18541, 17551, 199999, 3, 3, 18, 87037, 1108909, 157141, 2, 154981, 9, 1483333

Offset: 1

Marius A. Burtea, Sep 18 2020

m is a Moran number if m /digsum(m) is a prime number (A001101).

a(n) = 1 if and only if n is a Moran number.

			For n = 6, (1*6) / digsum(1*6) = 1, (2*6) / digsum(2*6) = 12 / 3 = 4, (3*6) / digsum(3*6) = 18 / 9 = 2 = prime(1), so a(6) = 3.
For n = 7, (1*7) / digsum(1*7) = 1, (2*7) / digsum(2*7) = 14 / 5, (3*7) / digsum(3*7) = 21 / 3 = 7 = prime(4), so a(7) = 3.

Cf. A001101, A007953, A144261.

moran:=func;
a:=[]; for n in [1..60] do k:=1; while not moran(k*n) do k:=k+1; end while; Append(~a,k); end for; a;

moranQ[n_] := PrimeQ[n / Plus @@ IntegerDigits[n]]; a[n_] := Module[{k = 1}, While[!moranQ[k*n], k++]; k]; Array[a, 60] (* Amiram Eldar, Sep 19 2020 *)

cp's OEIS Frontend

A144364 Where records occur in A144261.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A144363 Records in A144261.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Extensions

A358067 a(n) is the smallest m such that A144261(m) = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A226169 Niven numbers when expressed in bases 1 through 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A385482 a(n) is the least number k such that k*n is a binary Niven number (A049445).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A226171 Smallest base in which n is not Niven (or zero if n is Niven in every base).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A357936 a(n) is the least multiple of n that is a Niven (or Harshad) number.