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A005349
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Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.
(Formerly M0481)
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307
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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)
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OFFSET
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1,2
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COMMENTS
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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
The asymptotic density of this sequence is 0 (Cooper and Kennedy, 1984). - Amiram Eldar, Jul 10 2020
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.
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)
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REFERENCES
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Paul Dahlenberg and T. Edgar, Consecutive factorial base Niven numbers, Fib. Q., 56:2 (2018), 163-166.
D. R. Kaprekar, Multidigital Numbers, Scripta Math., Vol. 21 (1955), p. 27.
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.
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.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 381.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 171.
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LINKS
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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]
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EXAMPLE
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195 is a term of the sequence because it is divisible by 15 (= 1 + 9 + 5).
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a005349 n = a005349_list !! (n-1)
a005349_list = filter ((== 0) . a070635) [1..]
(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
(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
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CROSSREFS
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Cf. A001101, A007602, A007953, A028834, A038186, A049445, A052018, A052019, A052020, A052021, A052022, A065877, A070635, A113315, A188641.
Cf. A118363 (for factorial-base analog).
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KEYWORD
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nonn,base,nice,easy
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AUTHOR
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STATUS
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approved
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