A049445 -id:A049445 - cp's OEIS frontend

A337078 The number of binary Niven numbers (A049445) not exceeding 2^n.

2, 3, 5, 8, 13, 21, 37, 65, 124, 232, 431, 760, 1424, 2575, 4772, 8932, 17033, 32225, 61764, 117897, 224944, 428155, 814294, 1547596, 2934212, 5572886, 10609364, 20237826, 38773350, 74609953, 144275968, 280018507, 545782822, 1064716523, 2081890937, 4068716054

Offset: 1

Amiram Eldar, Aug 14 2020

			a(1) = 2 since there are 2 binary Niven numbers not exceeding 2^1: 1 and 2.

Amiram Eldar, Table of n, a(n) for n = 1..400 (calculated using a binary version of _Hiroaki Yamanouchi_'s Python code at A140866)
Jean-Marie De Koninck, Nicolas Doyon and Imre Kátai, On the counting function for the Niven numbers, Acta Arithmetica, Vol. 106, No. 3 (2003), 265-275.

Cf. A049445, A140866.

binNivenQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; s = {}; c = 0; p = 2; Do[If[binNivenQ[n], c++]; If[n == p, AppendTo[s, c]; p *= 2], {n, 1, 2^20}]; s

a(n) ~ 2^(n+1)/n (De Koninck et al., 2003, consequence of Theorem 1).

A152567 Numbers k such that A049445(k) is odd.

Original entry on oeis.org

1, 11, 19, 24, 27, 33, 43, 51, 54, 68, 71, 74, 76, 83, 89, 90, 98, 101, 107, 117, 130, 135, 138, 144, 151, 153, 156, 163, 165, 178, 181, 188, 195, 199, 203, 205, 207, 212, 215, 226, 230, 235, 238, 244, 249, 251, 258, 267, 272, 278, 282, 285, 294, 298, 304, 305, 325, 327

Offset: 1

Roger L. Bagula, Dec 07 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005349, A049445, A144302.

Position[Select[Range[1, 1500], Divisible[#, DigitCount[#, 2, 1]] &], ?OddQ] // Flatten (* _Amiram Eldar, Aug 16 2020 *)

Edited by N. J. A. Sloane, Dec 07 2008

A353988 Numbers k such that Fibonacci(k) is a binary Niven number (A049445).

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 10, 12, 18, 24, 30, 36, 48, 56, 60, 100, 120, 144, 150, 168, 240, 270, 288, 300, 324, 330, 336, 360, 444, 540, 594, 600, 624, 720, 750, 840, 864, 896, 900, 936, 1080, 1152, 1200, 1210, 1360, 1404, 1632, 1720, 1921, 2028, 2400, 2520, 2552, 2864

Offset: 1

Amiram Eldar, May 13 2022

Numbers k such that A011373(k) | A000045(k).

			1 is a term since A000045(1) = A011373(1) = 1 and 1 | 1.
10 is a term since A000045(10) = 55, A011373(1) = 5 and 5 | 55.

Amiram Eldar, Table of n, a(n) for n = 1..3000

Cf. A000045, A000120, A011373, A049445, A117774, A337448 (decimal analog).

Select[Range[3000], Divisible[(f = Fibonacci[#]), DigitCount[f, 2, 1]] &]

isok(k) = my(f=fibonacci(k)); ! (f % hammingweight(f)); \\ Michel Marcus, May 13 2022

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

Original entry on oeis.org

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

A333426 Primorial base Niven numbers: numbers divisible by their sum of digits in primorial base (A276150).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 25, 30, 32, 33, 36, 40, 42, 44, 45, 48, 50, 60, 64, 65, 66, 68, 70, 72, 77, 84, 88, 90, 92, 96, 105, 108, 112, 117, 120, 132, 133, 136, 144, 150, 154, 156, 160, 168, 180, 182, 184, 189, 192, 198, 200, 210, 212, 213, 216, 220

Offset: 1

Amiram Eldar, Mar 20 2020

nonn

Numbers k for which A276086(k) is in A373852. - Antti Karttunen, Jun 22 2024

			1 is a term since A276150(1) = 1 divides 1;
2 is a term since A276150(2) = 1 divides 2;

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

Cf. A005349, A049345, A049445, A064150, A064438, A064481, A118363, A235168, A276150, A328208, A328212, A373834 (characteristic function)
Cf. A276086, A333427, A333428, A341433, A347496, A358977, A373833, A373852.
Positions of 0's in A373832.

max = 5; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; sumdig[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; Select[Range[nmax], Divisible[#, sumdig[#]] &]

isA333426 = A373834; \\ Antti Karttunen, Jun 22 2024

A331728 Negabinary-Niven numbers: numbers divisible by the sum of digits in their negabinary representation (A027615).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 33, 35, 36, 40, 42, 48, 50, 52, 54, 56, 57, 60, 62, 63, 64, 66, 68, 69, 72, 76, 78, 80, 81, 84, 88, 90, 91, 95, 96, 100, 102, 108, 110, 112, 114, 120, 124, 125, 126, 128, 129, 132, 136, 138, 140

Offset: 1

Amiram Eldar, Jan 27 2020

			6 is a term since A039724(6) = 11010 and 1 + 1 + 0 + 1 + 0 = 3 is a divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A027615, A039724, A005349, A049445, A328208, A328212, A331085, A331088.

negaBinWt[n_] := negaBinWt[n] = If[n==0, 0, negaBinWt[Quotient[n-1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[n]]; Select[Range[100], negaBinNivenQ]

A064150 Numbers divisible by the sum of their ternary digits.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 45, 48, 54, 56, 57, 60, 63, 64, 65, 72, 75, 77, 78, 80, 81, 82, 84, 87, 88, 90, 92, 93, 95, 96, 99, 100, 105, 108, 111, 112, 115, 117, 120, 132, 133, 135, 136, 144, 145, 150, 152

Offset: 1

Len Smiley, Sep 11 2001

a(n) mod A053735(a(n)) = 0. - Reinhard Zumkeller, Nov 25 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith).
Paul Dahlenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quarterly, Vol. 56, No. 2 (2018), pp. 163-166.

Cf. A005349 (Decimal), A049445 (Binary).

a064150 n = a064150_list !! (n-1)
a064150_list = filter (\x -> x `mod` a053735 x == 0) [1..]
-- Reinhard Zumkeller, Oct 28 2012

Select[Range[200], IntegerQ[#/(Plus@@IntegerDigits[#, 3])] &] (* Alonso del Arte, May 27 2011 *)

isok(m)={m % sumdigits(m, 3) == 0} \\ Harry J. Smith, Sep 09 2009

import numpy as np
def gen():
    for dec_num in range(1,153):
        tern_num = np.base_repr(dec_num, 3)
        sum_tern_digits = 0
        for i in tern_num:
            sum_tern_digits += int(i)
        if dec_num % sum_tern_digits == 0:
            yield dec_num
print(list((gen()))) # Adrienne Leonardo, Dec 28 2024

Corrected and extended by Vladeta Jovovic, Sep 22 2001

Offset corrected by Reinhard Zumkeller, Oct 28 2012

A064438 Numbers which are divisible by the sum of their quaternary digits.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 21, 24, 28, 30, 32, 33, 35, 36, 40, 42, 48, 50, 52, 54, 60, 63, 64, 66, 68, 69, 72, 76, 78, 80, 81, 84, 88, 90, 91, 96, 100, 102, 108, 112, 114, 120, 126, 128, 129, 132, 136, 138, 140, 144, 148, 150, 154, 156, 160, 162, 168, 171, 180

Offset: 1

Len Smiley, Oct 01 2001

A good "puzzle" sequence -- guess the rule given the first twenty or so terms.

			Quaternary representation of 28 is 130, 1 + 3 + 0 = 4 divides 28.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Paul Dalenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quart. (2018) Vol. 56, No. 2, 163-166.

Cf. A005349 (decimal), A049445 (binary), A064150 (ternary).

maxarg := 190; for n := 1 to maxarg do if n mod sum(quaternarray(n)) = 0 then write(n," "); end; end; function quaternarray(n: integer): array; var k: integer; stk: stack; begin while n > 0 do k := n mod 4; stack_push(stk,k); n := (n - k) div 4; end; return stack2array(stk); end;

Select[Range[200],Divisible[#,Total[IntegerDigits[#,4]]]&] (* Harvey P. Dale, Jun 09 2011 *)

isok(n) = !(n % sumdigits(n, 4)); \\ Michel Marcus, Jun 24 2018

from sympy.ntheory.factor_ import digits
print([n for n in range(1, 201) if n%sum(digits(n, 4)[1:]) == 0]) # Indranil Ghosh, Apr 24 2017

More terms from Matthew Conroy, Oct 02 2001

Offset changed from 0 to 1 by Harry J. Smith, Sep 14 2009

A064481 Numbers which are divisible by the sum of their base-5 digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 26, 27, 28, 30, 32, 36, 40, 42, 45, 48, 50, 51, 52, 54, 56, 60, 63, 64, 65, 66, 72, 75, 76, 78, 80, 85, 88, 90, 91, 96, 99, 100, 102, 104, 105, 112, 117, 120, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144, 145

Offset: 1

Klaus Brockhaus, Oct 03 2001

			Base-5 representation of 28 is 103; 1 + 0 + 3 = 4 divides 28.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A005349 (base 10), A049445 (base 2), A064150 (base 3), A064438 (base 4), A344341.

: maxarg := 160; for n := 1 to maxarg do if n mod sum(basearray(n,5)) = 0 then write(n," "); end; end; function basearray(n,b: integer): array; var k: integer; stk: stack; begin while n > 0 do k := n mod b; stack_push(stk,k); n := (n - k) div b; end; return stack2array(stk); end;.

isok(n) = !(n % sumdigits(n, 5)); \\ Michel Marcus, Jun 24 2018

Offset changed from 0 to 1 by Harry J. Smith, Sep 15 2009

A334308 Base phi Niven numbers: numbers divisible by the number of 1's in their base phi representation (A055778).

Original entry on oeis.org

1, 2, 6, 12, 15, 16, 18, 20, 30, 35, 36, 45, 48, 55, 60, 70, 72, 78, 84, 90, 91, 95, 96, 98, 104, 108, 132, 144, 147, 154, 168, 175, 184, 189, 208, 224, 231, 232, 245, 252, 256, 261, 264, 270, 275, 280, 282, 287, 294, 315, 322, 324, 330, 336, 340, 342, 351, 357

Offset: 1

Amiram Eldar, Apr 22 2020

			6 is a term since its base phi representation is 1010.0001, and the number of 1's is 3, which is a divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Michel Dekking, The sum of digits function of the base phi expansion of the natural numbers, arXiv:1911.10705 [math.NT], 2019.
Ron Knott, Using Powers of Phi to represent Integers (Base Phi).
Wikipedia, Golden ratio base.

Cf. A005349, A049445, A055778, A064150, A064438, A064481, A118363, A328208, A328212, A333426.

phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n]] ][[1]]; Select[Range[360], Divisible[#, phiDigSum[#]] &]

cp's OEIS Frontend

A337078 The number of binary Niven numbers (A049445) not exceeding 2^n.

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A152567 Numbers k such that A049445(k) is odd.

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A353988 Numbers k such that Fibonacci(k) is a binary Niven number (A049445).

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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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A333426 Primorial base Niven numbers: numbers divisible by their sum of digits in primorial base (A276150).

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A331728 Negabinary-Niven numbers: numbers divisible by the sum of digits in their negabinary representation (A027615).

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A064150 Numbers divisible by the sum of their ternary digits.

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