A005349 - OEIS

%I M0481 #175 Nov 26 2024 13:39:16

%S 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,

%T 70,72,80,81,84,90,100,102,108,110,111,112,114,117,120,126,132,133,

%U 135,140,144,150,152,153,156,162,171,180,190,192,195,198,200,201,204

%N Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.

%C 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

%C 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

%C A070635(a(n)) = 0. A038186 is a subsequence. - _Reinhard Zumkeller_, Mar 10 2008

%C A049445 is a subsequence of this sequence. - _Ctibor O. Zizka_, Sep 06 2010

%C Complement of A065877; A188641(a(n)) = 1;  A070635(a(n)) = 0. - _Reinhard Zumkeller_, Apr 07 2011

%C A001101, the Moran numbers, are a subsequence. - _Reinhard Zumkeller_, Jun 16 2011

%C A140866 gives the number of terms <= 10^k. - _Robert G. Wilson v_, Oct 16 2012

%C The asymptotic density of this sequence is 0 (Cooper and Kennedy, 1984). - _Amiram Eldar_, Jul 10 2020

%C From _Amiram Eldar_, Oct 02 2023: (Start)

%C Named "Harshad numbers" by the Indian recreational mathematician Dattatreya Ramchandra Kaprekar (1905-1986) in 1955. The meaning of the word is "giving joy" in Sanskrit.

%C Named "Niven numbers" by Kennedy et al. (1980) after the Canadian-American mathematician Ivan Morton Niven (1915-1999). During a lecture given at the 5th Annual Miami University Conference on Number Theory in 1977, Niven mentioned a question of finding a number that equals twice the sum of its digits, which appeared in the children's pages of a newspaper. (End)

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

%D D. R. Kaprekar, Multidigital Numbers, Scripta Math., Vol. 21 (1955), p. 27.

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

%D Robert E. Kennedy, Terry A. Goodman, and Clarence H. Best, Mathematical Discovery and Niven Numbers, The MATYC Journal, Vol. 14, No. 1 (1980), pp. 21-25.

%D József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 381.

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

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

%H N. J. A. Sloane, <a href="/A005349/b005349.txt">Table of n, a(n) for n = 1..11872</a> (all a(n) <= 100000)

%H Bob Albrecht, Don Albers, and Jim Conlan, <a href="https://archive.org/details/1980-07-recreational-computing/page/n57">Problem #22 Two-digit Niven numbers</a>, Programming Problems, Recreational Computing Magazine, Vol. 9, No. 1, Issue 46 (1980), p. 59.

%H Curtis N. Cooper and Robert E. Kennedy, <a href="https://doi.org/10.1155/S0161171285000576">On an asymptotic formula for the Niven numbers</a>, International Journal of Mathematics and Mathematical Sciences, Vol. 8, No. 3 (1985), pp. 537-543.

%H Curtis N. Cooper and Robert E. Kennedy, <a href="http://www.jstor.org/stable/2323194">Chebyshev's inequality and natural density</a>, Amer. Math. Monthly 96 (1989), no. 2, 118-124.

%H Paul Dalenberg and Tom Edgar, <a href="https://www.fq.math.ca/56-2.html">Consecutive factorial base Niven numbers</a>, Fibonacci Quart. (2018) Vol. 56, No. 2, 163-166.

%H Jean-Marie De Koninck and Nicolas Doyon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Doyon/doyon.html">Large and Small Gaps Between Consecutive Niven Numbers</a>, J. Integer Seqs., Vol. 6, 2003, Article 03.2.5.

%H Nicolas Doyon, <a href="https://www.collectionscanada.gc.ca/obj/s4/f2/dsk3/QQLA/TC-QQLA-24023.pdf">Les fascinants nombres de Niven</a>, Thèse de la faculté des sciences et de génie de l'université Laval, Québec,‎ Novembre 2006 (in French).

%H Ömer Eğecioğlu and Bünyamin Şahin, <a href="https://toc.ui.ac.ir/article_28781.html">On twin EP numbers</a>, Transact. Comb. (2025) Vol. 14, Iss. 4, Art. No. 4, 261-270. See p. 262.

%H Richard K. Guy, <a href="http://www.jstor.org/stable/2691503">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20.

%H Richard K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]

%H Robert E. Kennedy, <a href="https://cms.math.ca/wp-content/uploads/crux-pdfs/Crux_v8n05_May.pdf">Digital sums, Niven numbers, and natural density</a>, Crux Mathematicorum, Vol. 8 (1982), pp. 131-135.

%H Robert E. Kennedy and Curtis N. Cooper, <a href="http://www.jstor.org/stable/2686395">On the natural density of the Niven numbers</a>, The College Mathematics Journal, Vol. 15, No. 4 (Sep., 1984), pp. 309-312.

%H Project Euler, <a href="https://projecteuler.net/problem=387">Harshad Numbers: Problem 387</a>.

%H Terry Trotter, <a href="https://web.archive.org/web/20051108143157/http://www.trottermath.net/numthry/nivennos.html">Niven Numbers for Fun and Profit</a>. [archived page]

%H Gérard Villemin, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Decompos/Harshad.htm">Nombres de Harshad</a> (French).

%H Elaine E. Visitacion, Renalyn T. Boado, Mary Ann V. Doria, and Eduard M. Albay, <a href="https://web.archive.org/web/20190407151217/http://www.dmmmsu-sluc.com/wp-content/uploads/2018/03/CAS-Monitor-2016-2017-1.pdf">On Harshad Number</a>, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 134-138. [archived]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Digit.html">Digit</a> and <a href="http://mathworld.wolfram.com/HarshadNumber.html">Harshad Numbers</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Harshad_number">Harshad number</a>.

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

%p 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_

%t 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 *)

%t Select[Range[300],Divisible[#,Total[IntegerDigits[#]]]&] (* _Harvey P. Dale_, Sep 07 2015 *)

%o (Haskell)

%o a005349 n = a005349_list !! (n-1)

%o a005349_list = filter ((== 0) . a070635) [1..]

%o -- _Reinhard Zumkeller_, Aug 17 2011, Apr 07 2011

%o (Magma) [n: n in [1..250] | n mod &+Intseq(n) eq 0];  // _Bruno Berselli_, May 28 2011

%o (Magma) [n: n in [1..250] | IsIntegral(n/&+Intseq(n))];  // _Bruno Berselli_, Feb 09 2016

%o (PARI) is(n)=n%sumdigits(n)==0 \\ _Charles R Greathouse IV_, Oct 16 2012

%o (Python)

%o 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

%o (Sage)

%o [n for n in (1..10^4) if sum(n.digits(base=10)).divides(n)] # _Freddy Barrera_, Jul 27 2018

%o (GAP) Filtered([1..230],n-> n mod List(List([1..n],ListOfDigits),Sum)[n]=0); # _Muniru A Asiru_

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

%Y Cf. A001102 (a subsequence).

%Y Cf. A118363 (for factorial-base analog).

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

%K nonn,base,nice,easy

%O 1,2

%A _N. J. A. Sloane_, _Robert G. Wilson v_