A005349 Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits. (Formerly M0481)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 195, 198, 200, 201, 204

Offset

1,2

Comments

Both spellings, "Harshad" or "harshad", are in use. It is a Sanskrit word, and in Sanskrit there is no distinction between upper- and lower-case letters. - N. J. A. Sloane, Jan 04 2022

z-Niven numbers are numbers n which are divisible by (A*s(n) + B) where A, B are integers and s(n) is sum of digits of n. Niven numbers have A = 1, B = 0. - Ctibor O. Zizka, Feb 23 2008

A070635(a(n)) = 0. A038186 is a subsequence. - Reinhard Zumkeller, Mar 10 2008

A049445 is a subsequence of this sequence. - Ctibor O. Zizka, Sep 06 2010

Complement of A065877; A188641(a(n)) = 1; A070635(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2011

A001101, the Moran numbers, are a subsequence. - Reinhard Zumkeller, Jun 16 2011

A140866 gives the number of terms <= 10^k. - Robert G. Wilson v, Oct 16 2012

The asymptotic density of this sequence is 0 (Cooper and Kennedy, 1984). - Amiram Eldar, Jul 10 2020

References

Paul Dahlenberg and T. Edgar, Consecutive factorial base Niven numbers, Fib. Q., 56:2 (2018), 163-166.

R. E. Kennedy and C. N. Cooper, On the natural density of the Niven numbers, Abstract 816-11-219, Abstracts Amer. Math. Soc., 6 (1985), 17.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 171.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..11872 (all a(n) <= 100000)

C. N. Cooper and R. E. Kennedy, Chebyshev's inequality and natural density, Amer. Math. Monthly 96 (1989), no. 2, 118-124.

Paul Dalenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quart. (2018) Vol. 56, No. 2, 163-166.

Jean-Marie De Koninck and Nicolas Doyon, Large and Small Gaps Between Consecutive Niven Numbers, J. Integer Seqs., Vol. 6, 2003, Article 03.2.5.

R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.

R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]

R. E. Kennedy and C. N. Cooper, On the natural density of the Niven numbers, The College Mathematics Journal, Vol. 15, No. 4 (Sep., 1984), pp. 309-312.

Project Euler, Harshad Numbers: Problem 387

Terry Trotter, Niven Numbers for Fun and Profit [archived page]

Gérard Villemin, Nombres de Harshad (French)

Elaine E. Visitacion, Renalyn T. Boado, Mary Ann V. Doria, and Eduard M. Albay, On Harshad Number, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 134-138. [archived]

Eric Weisstein's World of Mathematics, Digit and Harshad Numbers

Wikipedia, Harshad number

Example

195 is a term of the sequence because it is divisible by 15 (= 1 + 9 + 5).

Maple

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: seq(p(n), n=1..210); # Emeric Deutsch

Mathematica

harshadQ[n_] := Mod[n, Plus @@ IntegerDigits@ n] == 0; Select[ Range[1000], harshadQ] (* Alonso del Arte, Aug 04 2004 and modified by Robert G. Wilson v, Oct 16 2012 *)
Select[Range[300], Divisible[#, Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Sep 07 2015 *)

Prog

(Haskell)
a005349 n = a005349_list !! (n-1)
a005349_list = filter ((== 0) . a070635) [1..]
-- Reinhard Zumkeller, Aug 17 2011, Apr 07 2011
(Magma) [n: n in [1..250] | n mod &+Intseq(n) eq 0];  // Bruno Berselli, May 28 2011
(Magma) [n: n in [1..250] | IsIntegral(n/&+Intseq(n))];  // Bruno Berselli, Feb 09 2016
(PARI) is(n)=n%sumdigits(n)==0 \\ Charles R Greathouse IV, Oct 16 2012
(Python)
A005349 = [n for n in range(1, 10**6) if not n % sum([int(d) for d in str(n)])] # Chai Wah Wu, Aug 22 2014
(Sage)
[n for n in (1..10^4) if sum(n.digits(base=10)).divides(n)] # Freddy Barrera, Jul 27 2018
(GAP) Filtered([1..230], n-> n mod List(List([1..n], ListOfDigits), Sum)[n]=0); # Muniru A Asiru

Crossrefs

Cf. A001101, A007602, A007953, A028834, A038186, A049445, A052018, A052019, A052020, A052021, A052022, A065877, A070635, A113315, A188641.

Cf. A001102 (a subsequence).

Cf. A118363 (for factorial-base analog).

Cf. A330927, A154701, A141769, A330928, A330929, A330930 (start of runs of 2, 3, ..., 7 consecutive Niven numbers).

Sequence in context: A064807 A235591 A007603 * A234474 A285829 A225780

Adjacent sequences: A005346 A005347 A005348 * A005350 A005351 A005352

Keyword

nonn,base,nice,easy

Author

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

Status

approved