A005349 -id:A005349 - cp's OEIS frontend

A342262 Numbers divisible both by the product of their nonzero digits (A055471) and by the sum of their digits (A005349).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 24, 30, 36, 40, 50, 60, 70, 80, 90, 100, 102, 110, 111, 112, 120, 132, 135, 140, 144, 150, 200, 210, 216, 220, 224, 240, 300, 306, 312, 315, 360, 400, 432, 480, 500, 510, 540, 550, 600, 612, 624, 630, 700, 735, 800, 900, 1000, 1002, 1008

Offset: 1

Bernard Schott, Mar 27 2021

Equivalently, Niven numbers that are divisible by the product of their nonzero digits. A Niven number (A005349) is a number that is divisible by the sum of its digits.
Niven numbers without zero digit that are divisible by the product of their digits are in A038186.
Differs from super Niven numbers, the first 16 terms are the same, then A328273(17) = 48 while a(17) = 50.
This sequence is infinite since if m is a term, then 10*m is another term.

			The product of the nonzero digits of 306 =  3*6 = 18, and 306 divided by 18 = 17. The sum of the digits of 306 = 3 + 0 + 6 = 9, and 306 divided by 9 = 34. Thus 306 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Intersection of A005349 and A055471.
Supersequence of A038186.
Cf. A051004, A328273, A342650.

q[n_] := And @@ Divisible[n, {Times @@ (d = Select[IntegerDigits[n], # > 0 &]), Plus @@ d}]; Select[Range[1000], q] (* Amiram Eldar, Mar 27 2021 *)
Select[Range[1200],Mod[#,Times@@(IntegerDigits[#]/.(0->1))]== Mod[#,Total[ IntegerDigits[#]]]==0&] (* Harvey P. Dale, Sep 26 2021 *)

isok(m) = my(d=select(x->(x!=0), digits(m))); !(m % vecprod(d)) && !(m % vecsum(d)); \\ Michel Marcus, Mar 27 2021

Example clarified by Harvey P. Dale, Sep 26 2021

A082517 Numbers n such that N(n+1) - N(n) sets a new record, where N(n) = A005349.

Original entry on oeis.org

1, 10, 11, 26, 28, 32, 83, 102, 143, 561, 716, 1118, 2948, 4194, 5439, 33494, 51544, 61588, 94748, 265336, 800054, 3750017, 6292149, 44194186, 55065654, 61074615, 179838772, 399977785, 497993710, 502602764, 547594831, 1039028518, 14192408715, 14761794180

Offset: 1

Jason Earls, Apr 30 2003

Where records occur in A082516. Cf. A005349.

a(22)-a(34) from Donovan Johnson, Nov 10 2011

A092787 Primes in the sequence A005349 - 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 41, 47, 53, 59, 71, 79, 83, 89, 101, 107, 109, 113, 131, 139, 149, 151, 179, 191, 197, 199, 223, 227, 229, 233, 239, 251, 263, 269, 307, 311, 359, 401, 409, 419, 431, 439, 443, 449, 467, 479, 499, 503, 509, 521, 557, 587, 593, 599

Offset: 1

Jorge Coveiro, Apr 14 2004

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

Cf. A005349.

nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; Select[Range[600], PrimeQ[#] && nivenQ[# + 1] &] (* Amiram Eldar, Dec 05 2021 *)

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 24 2004

A100432 Bisection of A005349.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 20, 24, 30, 40, 45, 50, 60, 70, 80, 84, 100, 108, 111, 114, 120, 132, 135, 144, 152, 156, 171, 190, 195, 200, 204, 209, 216, 222, 225, 230, 240, 247, 261, 266, 280, 288, 306, 312, 320, 324, 333, 342, 360, 370, 375, 392, 399, 402, 407, 410, 420

Offset: 1

N. J. A. Sloane, Nov 20 2004

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A005349, A100433.

A005349:=[n: n in [1..10000] | n mod &+Intseq(n) eq 0];
A100432:= func< n | A005349[2*n-1] >;
[A100432(n): n in [1..150]]; // G. C. Greubel, Apr 09 2023

s:=proc(n) local N:N:=convert(n,base,10):sum(N[j],j=1..nops(N)) end:p:=proc(n) if floor(n/s(n))=n/s(n) then n else fi end: A:=[seq(p(n),n=1..440)]: seq(A[2*j-1],j=1..58); # Emeric Deutsch, Dec 16 2004

Select[Range[1000], Divisible[#, Total[IntegerDigits[#]]] &][[1;; ;; 2]] (* G. C. Greubel, Apr 09 2023 *)

A005349=[n for n in (1..10^4) if sum(n.digits(base=10)).divides(n)]
def A100432(n): return A005349[2*n-2]
[A100432(n) for n in range(1, 151)] # G. C. Greubel, Apr 09 2023

a(n) = A005349(2*n-1). - G. C. Greubel, Apr 09 2023

More terms from Emeric Deutsch, Dec 16 2004

A349484 Niven numbers whose arithmetic derivative is also a Niven number (A005349).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 18, 20, 21, 27, 36, 48, 50, 54, 72, 81, 100, 108, 111, 112, 135, 153, 156, 180, 192, 201, 209, 210, 216, 224, 225, 230, 243, 280, 288, 306, 324, 336, 351, 364, 378, 392, 400, 405, 407, 420, 432, 441, 480, 481, 486, 500, 504, 511, 512

Offset: 1

Marius A. Burtea, Nov 20 2021

The sequence is infinite because the numbers of the form m = 2*10^(10^k), k >= 1, are terms. Indeed, m is a Niven number, m' = 10^(10^k) + 2*10^k*10^(10^k - 1)*7 = 10^(10^k - 1)*(10 + 140*10^k) = 10^(10^k)*(1 + 14*10^k), digsum(m') = 6 and m' is divisible by 6, so it is a Niven number.

			2 = A005349(2) and 2' = 1 = A005349(1), so 2 is a term.
18 = A005349(12) and 18' = 21 = A005349(14), so 18 is a term.

Cf. A002808, A005349 (Niven numbers), A003415 (arithmetic derivative).

f:=func; a:=[]; niven:=func; [n:n in [2..520]|niven(n) and niven(Floor(f(n)))];

nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[2, 512], And @@ nivenQ /@ {#, d[#]} &] (* Amiram Eldar, Nov 20 2021 *)

A007602 Numbers that are divisible by the product of their digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 36, 111, 112, 115, 128, 132, 135, 144, 175, 212, 216, 224, 312, 315, 384, 432, 612, 624, 672, 735, 816, 1111, 1112, 1113, 1115, 1116, 1131, 1176, 1184, 1197, 1212, 1296, 1311, 1332, 1344, 1416, 1575, 1715, 2112, 2144

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

These are called Zuckerman numbers to base 10. [So-named by J. J. Tattersall, after Herbert S. Zuckerman. - Charles R Greathouse IV, Jun 06 2017] - Howard Berman (howard_berman(AT)hotmail.com), Nov 09 2008
This sequence is a subsequence of A180484; the first member of A180484 that is not a member of A007602 is 1114. - D. S. McNeil, Sep 09 2010
Complement of A188643; A188642(a(n)) = 1; A038186 is a subsequence; A168046(a(n)) = 1: subsequence of A052382. - Reinhard Zumkeller, Apr 07 2011
The terms of n digits in the sequence, for n from 1 to 14, are 9, 5, 20, 40, 117, 285, 747, 1951, 5229, 13493, 35009, 91792, 239791, 628412, 1643144, 4314987. Empirically, the counts seem to grow as 0.858*2.62326^n. - Giovanni Resta, Jun 25 2017
De Koninck and Luca showed that the number of Zuckerman numbers below x is at least x^0.122 but at most x^0.863. - Tomohiro Yamada, Nov 17 2017
The quotients obtained when Zuckerman numbers are divided by the product of their digits are in A288069. - Bernard Schott, Mar 28 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters (2005), 2nd Edition, p. 86 (see problems 44-45).

Reinhard Zumkeller and Zak Seidov, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Florian Luca, Positive integers divisible by the product of their nonzero digits, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below)
Jean-Marie De Koninck and Florian Luca, Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85, Port. Math. 74 (2017), 169-170.
Qizheng He and Carlo Sanna, Counting numbers that are divisible by the product of their digits, arXiv:2403.14812 [math.NT], 2024.
Giovanni Resta, Zuckerman numbers, Numbers Aplenty.
Index entries for Colombian or self numbers and related sequences

Cf. A034709, A001103, A005349, A288069.
Cf. A286590 (for factorial-base analog).
Subsequence of A002796, A034838, and A055471.

import Data.List (elemIndices)
a007602 n = a007602_list !! (n-1)
a007602_list = map succ $ elemIndices 1 $ map a188642 [1..]
-- Reinhard Zumkeller, Apr 07 2011

[ n: n in [1..2144] | not IsZero(&*Intseq(n)) and IsZero(n mod &*Intseq(n)) ];  // Bruno Berselli, May 28 2011

filter:= proc(n)
local p;
p:= convert(convert(n,base,10),`*`);
p <> 0 and n mod p = 0
end proc;
select(filter, [$1..10000]); # Robert Israel, Aug 24 2014

zuckerQ[n_] := Module[{d = IntegerDigits[n], prod}, prod = Times @@ d; prod > 0 && Mod[n, prod] == 0]; Select[Range[5000], zuckerQ] (* Alonso del Arte, Aug 04 2004 *)

for(n=1,10^5,d=digits(n);p=prod(i=1,#d,d[i]);if(p&&n%p==0,print1(n,", "))) \\ Derek Orr, Aug 25 2014

from operator import mul
from functools import reduce
A007602 = [n for n in range(1,10**5) if not (str(n).count('0') or n % reduce(mul, (int(d) for d in str(n))))] # Chai Wah Wu, Aug 25 2014

A049445 Numbers k with the property that the number of 1's in binary expansion of k (see A000120) divides k.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 21, 24, 32, 34, 36, 40, 42, 48, 55, 60, 64, 66, 68, 69, 72, 80, 81, 84, 92, 96, 108, 110, 115, 116, 120, 126, 128, 130, 132, 136, 138, 144, 155, 156, 160, 162, 168, 172, 180, 184, 185, 192, 204, 205, 212, 216, 220, 222, 228

Offset: 1

N. J. A. Sloane

If instead of base 2 we take base 10, then we have the so-called Harshad or Niven numbers (i.e., positive integers divisible by the sum of their digits; A005349). - Emeric Deutsch, Apr 11 2007

A199238(a(n)) = 0. - Reinhard Zumkeller, Nov 04 2011

			20 is in the sequence because 20 is written 10100 in binary and 1 + 1 = 2, which divides 20.
21 is in the sequence because 21 is written 10101 in binary and 1 + 1 + 1 = 3, which divides 21.
22 is not in the sequence because 22 is written 10110 in binary 1 + 1 + 1 = 3, which does not divide 22.

Indranil Ghosh, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Paul Dalenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quarterly, Vol. 56, No. 2 (2018), pp. 163-166.
Jean-Marie De Koninck, Nicolas Doyon and Imre Kátai, On the counting function for the Niven numbers, Acta Arithmetica, Vol. 106, No. 3 (2003), pp. 265-275.

Cf. A000120, A005349, A199238.

a049445 n = a049445_list !! (n-1)
a049445_list = map (+ 1) $ elemIndices 0 a199238_list
-- Reinhard Zumkeller, Nov 04 2011

a:=proc(n) local n2: n2:=convert(n,base,2): if n mod add(n2[i],i=1..nops(n2)) = 0 then n else fi end: seq(a(n),n=1..300); # Emeric Deutsch, Apr 11 2007

binHarshadQ[n_] := Divisible[n, Count[IntegerDigits[n, 2], 1]]; Select[Range[228], binHarshadQ] (* Jean-François Alcover, Dec 01 2011 *)
Select[Range[300],Divisible[#,DigitCount[#,2,1]]&] (* Harvey P. Dale, Mar 20 2016 *)

for(n=1,1000,b=binary(n);l=length(b); if(n%sum(i=1,l, component(b,i))==0,print1(n,",")))

is_A049445(n)={n%norml2(binary(n))==0} \\ M. F. Hasler, Oct 09 2012

isok(n) = ! (n % hammingweight(n)); \\ Michel Marcus, Feb 10 2016

A049445 = [n for n in range(1,10**5) if not n % sum([int(d) for d in bin(n)[2:]])] # Chai Wah Wu, Aug 22 2014

{k : A000120(k) | k}. - R. J. Mathar, Mar 03 2008
a(n) seems to be asymptotic to c*n*log(n) where 0.7 < c < 0.8. - Benoit Cloitre, Jan 22 2003
Heuristically, c should be 1/(2*log(2)), since a random d-bit number should have probability approximately 2/d of being in the sequence. - Robert Israel, Aug 22 2014
{a(n)} = {k : A199238(k) = 0}. - M. F. Hasler, Oct 09 2012
De Koninck et al. (2003) proved that the number of base-b Niven numbers not exceeding x, N_b(x), is asymptotically equal to ((2*log(b)/(b-1)^2) * Sum_{j=1..b-1} gcd(j, b-1) + o(1)) * x/log(x). For b=2, N_2(n) ~ (2*log(2) + o(1)) * x/log(x). Therefore, the constant c mentioned above is indeed 1/(2*log(2)). - Amiram Eldar, Aug 16 2020

More terms from Michael Somos

Edited by N. J. A. Sloane, Oct 07 2005 and May 16 2008

A328208 Zeckendorf-Niven numbers: numbers divisible by the number of terms in their Zeckendorf representation (A007895).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 16, 18, 21, 22, 24, 26, 27, 30, 34, 36, 42, 45, 48, 55, 56, 58, 60, 66, 68, 69, 72, 76, 78, 80, 81, 84, 89, 90, 92, 93, 94, 96, 99, 102, 105, 108, 110, 111, 116, 120, 126, 132, 135, 140, 144, 146, 150, 152, 153, 156, 159, 162

Offset: 1

Amiram Eldar, Oct 07 2019

nonn

			12 is in the sequence since A007895(12) = 3 and 3 is a divisor of 12.

Andrew Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.

Robert Israel, Table of n, a(n) for n = 1..10000
Helen G. Grundman, Consecutive Zeckendorf-Niven and lazy-Fibonacci-Niven numbers, Fibonacci Quarterly, Vol. 45, No. 3 (2007), pp. 272-276.
Andrew Ray and Curtis Cooper, On the natural density of the k-Zeckendorf Niven numbers, J. Inst. Math. Comput. Sci. Math., Vol. 19 (2006), pp. 83-98.

Cf. A005349, A007895.

fib:= combinat:-fibonacci:
phi:= 1/2 + sqrt(5)/2:
fibapp:= n -> phi^n/sqrt(5):
invfib := proc(x::posint)
  local q, n;
  q:= evalf((ln(x+1/2) + ln(5)/2)/ln(phi));
  n:= floor(q);
  if fib(n) <= x then
    while fib(n+1) <= x do
      n := n+1
    end do
  else
    while fib(n) > x do
      n := n-1
    end do
  end if;
  n
end proc:
zeck:= proc(x) local n;
 if x = 0 then 0
 else
   n:= invfib(x);
   F[n] + zeck(x-fib(n));
 fi
end proc:
filter:= n -> n mod nops(zeck(n)) = 0:
select(filter, [$1..200]); # Robert Israel, Oct 25 2019

z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; aQ[n_] := Divisible[n, z[n]]; Select[Range[1000], aQ] (* after Alonso del Arte at A007895 *)

A118363 Factorial base Niven (or Harshad) numbers: numbers that are divisible by the sum of their factorial base digits.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 26, 27, 30, 35, 36, 40, 48, 52, 54, 56, 60, 70, 72, 75, 80, 90, 91, 96, 105, 108, 112, 117, 120, 122, 123, 126, 132, 135, 140, 144, 148, 150, 152, 156, 161, 168, 175, 180, 186, 192, 204, 208, 210, 222, 224, 240, 244, 245, 246

Offset: 1

Alonso del Arte, May 15 2006

Also called "Fiven" numbers [Dahlenberg and Edgar]. - N. J. A. Sloane, Jun 25 2018

			a(8) = 16 because it is written 220 in factorial base and 2 + 2 + 0 = 4, which is a divisor of 16.
17 is not on the list because it is written 221 in factorial base and 2 + 2 + 1 = 5, which is not a divisor of 17.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Paul Dahlenberg and Tom Edgar, Consecutive factorial base Niven numbers, The Fibonacci Quarterly, Vol. 56, No. 2 (2018), pp. 163-166.
Index entries for sequences related to factorial base representation.

Cf. A007623 (integers written in factorial base), A005349 (base 10 Harshad numbers).
Cf. A286607 (complement), A034968, A286590.
Positions of zeros in A286604.

(*For the definition of the factorial base version of IntegerDigits, see A007623*) Select[Range[250],IntegerQ[ #/(Plus@@factBaseIntDs[ # ])]&]

is(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); !(n % s);} \\ Amiram Eldar, Oct 08 2024

def a007623(n, p=2): return n if nIndranil Ghosh, Jun 21 2017

A154701 Numbers k such that k, k + 1 and k + 2 are 3 consecutive Harshad numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 110, 510, 511, 1010, 1014, 1015, 2022, 2023, 2464, 3030, 3031, 4912, 5054, 5831, 7360, 8203, 9854, 10010, 10094, 10307, 10308, 11645, 12102, 12103, 12255, 12256, 13110, 13111, 13116, 13880, 14704, 15134, 17152, 17575, 18238, 19600, 19682

Offset: 1

Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 14 2009, Jan 15 2009

Harshad numbers are also known as Niven numbers.

Cooper and Kennedy proved that there are infinitely many runs of 20 consecutive Niven numbers. Therefore this sequence is infinite. - Amiram Eldar, Jan 03 2020

			110 is a term since 110 is divisible by 1 + 1 + 0 = 2, 111 is divisible by 1 + 1 + 1 = 3, and 112 is divisible by 1 + 1 + 2 = 4.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 36, entry 110.

Robert Israel, Table of n, a(n) for n = 1..10000
Curtis Cooper and Robert E. Kennedy, On consecutive Niven numbers, Fibonacci Quarterly, Vol. 21, No. 2 (1993), pp. 146-151.
Helen G. Grundman, Sequences of consecutive Niven numbers, Fibonacci Quarterly, Vol. 32, No. 2 (1994), pp. 174-175.
Wikipedia, Harshad number
Brad Wilson, Construction of 2n consecutive n-Niven numbers, Fibonacci Quarterly, Vol. 35, No. 2 (1997), pp. 122-128.

A subset of A005349.

Cf. A060159, A141769, A328210, A328214, A330927, A330928, A330929, A330930, A330932.

#include 
#include 
int is_harshad(int n){
  int i,j,count=0;
  i=n;
  while(i>0){
    count=count+i%10;
    i=i/10;
  }
  return n%count==0?1:0;
}
main(){
  int k;
  clrscr();
  for(k=1;k<=30000;k++)
    if(is_harshad(k)&&is_harshad(k+1)&&is_harshad(k+2))
      printf("%d,",k);
  getch();
  return 0;
}

f:=func; a:=[]; for k in [1..20000] do  if forall{m:m in [0..2]|f(k+m)} then Append(~a,k); end if; end for; a; // Marius A. Burtea, Jan 03 2020

Res:= NULL: count:= 0:
state:= 1:
L:= [1]:
for n from 2 while count < 100 do
  L[1]:=L[1]+1;
  for k from 1 while L[k]=10 do L[k]:= 0;
    if k = nops(L) then L:= [0$nops(L),1]; break
    else L[k+1]:= L[k+1]+1 fi
  od:
  s:= convert(L,`+`);
  if n mod s = 0 then
     state:= min(state+1,3);
     if state = 3 then count:= count+1; Res:= Res, n-2; fi
  else state:= 0
  fi
od:
Res; # Robert Israel, Feb 01 2019

nivenQ[n_] := Divisible[n, Total @ IntegerDigits[n]]; niv = nivenQ /@ Range[3]; seq = {}; Do[niv = Join[Rest[niv], {nivenQ[k]}]; If[And @@ niv, AppendTo[seq, k - 2]], {k, 3, 2*10^4}]; seq (* Amiram Eldar, Jan 03 2020 *)

from itertools import count, islice
def agen(): # generator of terms
    h1, h2, h3 = 1, 2, 3
    while True:
        if h3 - h1 == 2: yield h1
        h1, h2, h3 = h2, h3, next(k for k in count(h3+1) if k%sum(map(int, str(k))) == 0)
print(list(islice(agen(), 45))) # Michael S. Branicky, Mar 17 2024

cp's OEIS Frontend

A342262 Numbers divisible both by the product of their nonzero digits (A055471) and by the sum of their digits (A005349).

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A082517 Numbers n such that N(n+1) - N(n) sets a new record, where N(n) = A005349.

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A092787 Primes in the sequence A005349 - 1.

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A100432 Bisection of A005349.

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A349484 Niven numbers whose arithmetic derivative is also a Niven number (A005349).

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A007602 Numbers that are divisible by the product of their digits.

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A049445 Numbers k with the property that the number of 1's in binary expansion of k (see A000120) divides k.

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