A113315 -id:A113315 - cp's OEIS frontend

A066579 Erroneous version of A113315.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 2, 10, 7, 4, 3, 10, 4, 10, 7, 5, 4, 10, 6, 10, 7, 10, 8, 9, 7

Offset: 1

N. J. A. Sloane, Mar 11 2014

dead

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

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

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

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
From Amiram Eldar, Oct 02 2023: (Start)
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)

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

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.

N. J. A. Sloane, Table of n, a(n) for n = 1..11872 (all a(n) <= 100000)
Bob Albrecht, Don Albers, and Jim Conlan, Problem #22 Two-digit Niven numbers, Programming Problems, Recreational Computing Magazine, Vol. 9, No. 1, Issue 46 (1980), p. 59.
Curtis N. Cooper and Robert E. Kennedy, On an asymptotic formula for the Niven numbers, International Journal of Mathematics and Mathematical Sciences, Vol. 8, No. 3 (1985), pp. 537-543.
Curtis N. Cooper and Robert 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.
Nicolas Doyon, Les fascinants nombres de Niven, Thèse de la Faculté des Sciences et de Génie de l'Université Laval, Québec, Novembre 2006 (in French).
Ömer Eğecioğlu and Bünyamin Şahin, On twin EP numbers, Transact. Comb. (2025) Vol. 14, Iss. 4, Art. No. 4, 261-270. See p. 262.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Robert E. Kennedy, Digital sums, Niven numbers, and natural density, Crux Mathematicorum, Vol. 8 (1982), pp. 131-135.
Robert E. Kennedy and Curtis 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.

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).

Filtered([1..230],n-> n mod List(List([1..n],ListOfDigits),Sum)[n]=0); # Muniru A Asiru

a005349 n = a005349_list !! (n-1)
a005349_list = filter ((== 0) . a070635) [1..]
-- Reinhard Zumkeller, Aug 17 2011, Apr 07 2011

[n: n in [1..250] | n mod &+Intseq(n) eq 0];  // Bruno Berselli, May 28 2011

[n: n in [1..250] | IsIntegral(n/&+Intseq(n))];  // Bruno Berselli, Feb 09 2016

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

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 *)

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

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

[n for n in (1..10^4) if sum(n.digits(base=10)).divides(n)] # Freddy Barrera, Jul 27 2018

A334375 a(n) equals the n-th Moran number (A001101) divided by its sum of digits.

Original entry on oeis.org

2, 7, 3, 7, 5, 7, 7, 37, 19, 13, 19, 19, 17, 13, 19, 19, 13, 11, 67, 23, 19, 37, 19, 19, 29, 19, 19, 37, 37, 31, 19, 67, 37, 47, 37, 31, 37, 73, 43, 37, 59, 37, 31, 37, 67, 37, 43, 37, 79, 73, 61, 41, 43, 37, 89, 73, 67, 47, 37, 61, 53, 37, 337, 127, 113, 109

Offset: 1

Rémy Sigrist, Apr 25 2020

			For n = 42:
- A001101(42) = 555,
- A007953(555) = 15,
- hence a(42) = 555/15 = 37.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A001101, A007953, A113315.

for (n=1, 1090, if (n%(s=sumdigits(n))==0 && isprime(n/s), print1 (n/s", ")))

a(n) = A001101(n) / A007953(A001101(n)).

A356349 Primitive Niven numbers: terms of A005349 that are not ten times another term of A005349.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 18, 21, 24, 27, 36, 42, 45, 48, 54, 63, 72, 81, 84, 102, 108, 110, 111, 112, 114, 117, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 190, 192, 195, 198, 201, 204, 207, 209, 216, 220, 222, 224, 225, 228, 230, 234

Offset: 1

Bernard Schott and Rémy Sigrist, Oct 15 2022

A005349(k) belongs to this sequence iff A113315(k) is not a multiple of 10.
This sequence is infinite as it contains A133384 and A199682.
Each Niven number can be uniquely written as a(m)*10^z for some m > 0 and z >= 0.
This sequence contains numbers with k trailing zeros for any k >= 0; for example R(2^k) * 10^k (where R = A002275).

			190 is a term as 190 is a Niven number and 19 is not a Niven number.
192 is a term as 192 is a Niven number and 192 is not divisible by 10.

Index entries for sequences related to decimal expansion of n

Cf. A002275, A005349, A113315, A133384, A199682.

is(n, base=10) = my (s=sumdigits(n, base)); n%s==0 && (n%base || (n/base)%s)

def ok(n):
    sd = sum(map(int, str(n)))
    return sd and not n%sd and (n%10 or (n//10)%sd)
print([k for k in range(235) if ok(k)]) # Michael S. Branicky, Oct 16 2022

A359960 Smallest Niven (or Harshad) number (A005349) with exactly n distinct prime factors.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 30030, 690690, 14804790, 223092870, 8254436190, 200560490130, 8222980095330, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1987938667108592728530, 117288381359406970983270, 7858321551080267055879090

Offset: 0

Bernard Schott, Jan 20 2023

a(11) = 200560490130; a(13) = 304250263527210.
a(n) >= A002110(n) = prime(n)#.
Many terms are primorial numbers, see A360011.

			2310 = 2*3*5*7*11 is the smallest integer with 5 prime factors because it is a primorial number, as 2310 / (2+3+1+0) = 385, 2310 is a Niven number: a(5) = 2310.

Giovanni Resta, Harshad numbers.
Eric Weisstein's World of Mathematics, Distinct Prime Factors.

Cf. A001221, A002110, A005349, A113315, A360011.

Similar: A060319 (Fibonacci), A083002 (oblong), A359961 (Zuckerman).

a(n) = my(k=1); while ((k % sumdigits(k)) || (omega(k) != n), k++); k; \\ Michel Marcus, Jan 20 2023

omega_niven(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A && v%sumdigits(v) == 0, listput(list, v)), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q)); list); vecsort(Vec(f(1, 2, n)));
a(n) = if(n==0, return(1)); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_niven(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Jan 22 2023

a(8)-a(9) from Michel Marcus, Jan 20 2023

a(10)-a(19) from Daniel Suteu, Jan 22 2023

A066355 A055471(n)/(product of nonzero digits of A055471(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 11, 6, 3, 10, 3, 10, 2, 10, 10, 10, 10, 10, 10, 100, 101, 51, 26, 21, 110, 111, 56, 23, 60, 8, 22, 9, 35, 9, 30, 5, 100, 13, 105, 53, 18, 55, 14, 30, 25, 100, 17, 52, 21, 20, 4, 100, 18, 15, 100, 102, 52, 27, 22, 100, 51, 13, 35, 8, 100, 7, 100

Offset: 1

Amarnath Murthy, Dec 20 2001

			15 is the 13th term of A055471, hence a(13)=15/(1*5) =3.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A055471, A113315.

pd[n_] := Times @@ Select[IntegerDigits[n], # > 0 &];
Select[Table[n/pd[n], {n, 1000}], IntegerQ] (* Ray Chandler, Mar 11 2014 *)

Corrected and extended by Ray Chandler, Mar 11 2014

A325454 a(n) is the digit sum of the n-th Niven number (or Harshad number).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 3, 9, 2, 3, 6, 9, 3, 9, 4, 6, 9, 12, 5, 9, 6, 9, 7, 9, 8, 9, 12, 9, 1, 3, 9, 2, 3, 4, 6, 9, 3, 9, 6, 7, 9, 5, 9, 6, 8, 9, 12, 9, 9, 9, 10, 12, 15, 18, 2, 3, 6, 9, 11, 3, 9, 4, 6, 8, 9, 12, 5, 9, 6, 9, 13, 9, 9, 12, 14, 9, 10, 15

Offset: 1

Felix Fröhlich, Sep 06 2019

Cf. A005349, A007953, A113315.

is_a005349(n) = n%sumdigits(n)==0
terms(n) = my(i=0); for(k=1, oo, if(i>=n, break); if(is_a005349(k), print1(sumdigits(k), ", "); i++))
terms(80) \\ Print initial 80 terms

a(n) = A005349(n)/A113315(n).

a(n) = A007953(A005349(n)).

A386248 a(n) is the unique integer k such that A161792(n) = k^A000120(A161792(n)).

Original entry on oeis.org

1, 2, 4, 8, 3, 16, 32, 6, 64, 128, 12, 256, 512, 24, 3, 1024, 2048, 48, 4096, 8192, 96, 16384, 32768, 192, 6, 65536, 131072, 384, 262144, 524288, 768, 1048576, 2097152, 1536, 12, 4194304, 3, 8388608, 3072, 3, 16777216, 33554432, 6144, 67108864, 134217728

Offset: 1

Rémy Sigrist, Jul 16 2025

			For n = 11: A161792(11) = 144 = 12^2 = 12^A000120(144), so a(11) = 12.

Cf. A000120, A113315, A161792, A386249.

{ for (n = 1, 2^27, if (ispower(n, hammingweight(n), &r), print1 (r", "););); }

A260348 Numbers n such that n is divisible by (10^k - digitsum(n)), where k equals the number of digits of digitsum(n).

Original entry on oeis.org

5, 8, 9, 18, 21, 24, 26, 27, 36, 44, 45, 50, 54, 60, 62, 63, 72, 80, 81, 86, 90, 108, 116, 117, 126, 132, 134, 135, 140, 144, 152, 153, 162, 170, 171, 180, 200, 204, 206, 207, 210, 216, 224, 225, 230, 234, 240, 242, 243, 252, 260, 261, 264, 270, 306, 312, 314

Offset: 1

Pieter Post, Jul 23 2015

This sequence is infinite because all numbers with a digitsum equal to 9 are part of this sequence.

			a(1) = 5, because 5 divided by (10 - 5) equals 1.
a(7) = 26, because digitsum(26) = 8 and 26 divided by (10 - 8) equals 13.
a(20) = 86, the first member of this sequence where digitsum(n) >= 10. Digitsum(86) = 14, so k = 10^2 - 14 = 86, so 86 is a member of this sequence.

Pieter Post, Table of n, a(n) for n = 1..12089

Cf. A005349, A007953, A113315.

fQ[n_] := Block[{d = Total@ IntegerDigits@ n, k}, k = IntegerLength@ d;
  Divisible[n, 10^k - d]]; Select[Range@ 314, fQ] (* or *)
Select[Range@ 314, Divisible[#, (10^(Floor[Log[10, Total@ IntegerDigits@ #]] + 1) - Total@ IntegerDigits@ #)] &] (* Michael De Vlieger, Aug 05 2015 *)

isok(n)=my(sd = sumdigits(n), nsd = #digits(sd)); n % (10^nsd - sd) == 0; \\ Michel Marcus, Aug 05 2015

def sod(n,m):
    kk = 0
    while n > 0:
        kk= kk+(n%m)
        n =int(n//m)
    return kk
for c in range (1, 10**6):
    k=len(str(sod(c,10)))
    kl=10**k-sod(c,10)
    if c%kl==0:
        print (c)

A334417 a(n) is the palindrome equal to A334416(n) divided by its sum of digits.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 2, 7, 4, 3, 4, 7, 5, 4, 6, 7, 8, 9, 7, 55, 22, 11, 55, 22, 55, 22, 55, 55, 33, 55, 55, 55, 55, 44, 55, 55, 55, 55, 505, 88, 66, 202, 262, 77, 121, 88, 99, 181, 151, 121, 101, 88, 505, 424, 181, 121, 151, 181, 131, 343, 202, 181, 141

Offset: 1

Bernard Schott, Apr 29 2020

			A334416(10) = 12 whose sum of digits is 3; 12/3 = 4, so a(10) = 4.

Index entries for sequences related to palindromes

Cf. A007953, A113315, A334416.

Cf. A334375 (similar for primes).

Select[#/Plus @@ IntegerDigits[#] & /@ Range[3000], PalindromeQ] (* Amiram Eldar, Apr 29 2020 *)

isok(m) = iferr(my(d=digits(m/sumdigits(m))); d==Vecrev(d), E, 0);
apply(x->x/sumdigits(x), select(x->isok(x), [1..3000])) \\ Michel Marcus, Apr 29 2020

a(n) = A334416(n) / A007953(A334416(n)).

cp's OEIS Frontend

A066579 Erroneous version of A113315.

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

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A334375 a(n) equals the n-th Moran number (A001101) divided by its sum of digits.

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A356349 Primitive Niven numbers: terms of A005349 that are not ten times another term of A005349.

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A359960 Smallest Niven (or Harshad) number (A005349) with exactly n distinct prime factors.

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A066355 A055471(n)/(product of nonzero digits of A055471(n)).

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A325454 a(n) is the digit sum of the n-th Niven number (or Harshad number).

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A386248 a(n) is the unique integer k such that A161792(n) = k^A000120(A161792(n)).

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