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A005349 Niven (or Harshad) numbers: numbers that are divisible by the sum of their digits.
(Formerly M0481)
146
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 less or equal to 10^k. - Robert G. Wilson v, Oct 16 2012

REFERENCES

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

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. E. Kennedy, Niven Numbers for Fun and Profit

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

GĂ©rard Villemin, Nombres de Harshad (French)

Eric Weisstein's World of Mathematics, Harshad Numbers

Eric Weisstein's World of Mathematics, Digit

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); (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 *)

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..204] | n mod (&+Intseq(n)) eq 0 ];  // Bruno Berselli, May 28 2011

(PARI) is(n)=n%sumdigits(n)==0 \\ Charles R Greathouse IV, Oct 16 2012

CROSSREFS

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

A001102 is a subsequence.

Sequence in context: A064807 A235591 A007603 * A234474 A225780 A225782

Adjacent sequences:  A005346 A005347 A005348 * A005350 A005351 A005352

KEYWORD

nonn,base,nice,easy

AUTHOR

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

STATUS

approved

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Last modified August 23 01:19 EDT 2014. Contains 245976 sequences.