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 A005349 Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits. (Formerly M0481) 290
 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 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, Niven Numbers for Fun and Profit [Warning: As of March 2018 this site appears to have been hacked. Proceed with great caution. The original content should be retrieved from the Wayback machine and added here. - N. J. A. Sloane, Mar 29 2018] 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 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. 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, harshadQ] (* Alonso del Arte, Aug 04 2004 and modified by Robert G. Wilson v, Oct 16 2012 *) Select[Range, 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 STATUS approved

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Last modified September 26 22:02 EDT 2022. Contains 357051 sequences. (Running on oeis4.)