A118363 -id:A118363 - cp's OEIS frontend

A328205 Numbers m such that m and m+1 are consecutive factorial base Niven numbers (A118363).

1, 8, 26, 35, 90, 122, 244, 245, 300, 384, 440, 510, 722, 804, 844, 845, 935, 944, 984, 1014, 1079, 1224, 1232, 1444, 1445, 1518, 1584, 1589, 1727, 1728, 1736, 1770, 1880, 2159, 2184, 2232, 2240, 2528, 2540, 2650, 2820, 2980, 3032, 3263, 3640, 4199, 4328, 4848

Offset: 1

Amiram Eldar, Oct 07 2019

Dahlenberg & Edgar proved that this sequence is infinite.

			8 is in the sequence since both 8 and 9 are in A118363. A034968(8) = 2 is a divisor of 8 and A034968(9) = 3 is a divisor of 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Dahlenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quarterly, Vol. 56, No. 2 (2018), pp. 163-166; alternative link.

Cf. A007623, A034968, A118363.

sf[n_] := Module[{s = 0, i = 2, k = n}, While[k > 0, k = Floor[n/i!]; s = s + (i - 1)*k; i++]; n - s]; fnQ[n_] := Divisible[n, sf[n]]; aQ[n_] := AllTrue[n + Range[0, 1], fnQ]; Select[Range[5000], aQ] (* after Jean-François Alcover at A034968 *)

A328206 Starts of runs of 3 consecutive factorial base Niven numbers (A118363).

Original entry on oeis.org

244, 844, 1444, 1727, 5164, 5764, 5950, 10084, 10967, 13583, 15190, 20207, 21130, 22048, 40444, 40535, 41044, 45364, 45550, 56015, 60730, 62848, 63479, 80644, 91408, 132208, 153340, 163799, 173008, 176110, 178007, 195983, 242368, 280852, 283168, 363004, 363604

Offset: 1

Amiram Eldar, Oct 07 2019

Dahlenberg & Edgar proved that this sequence is infinite.

			244 is in the sequence since 244, 245 and 246 are in A118363. A034968(244) = 4 is a divisor of 244, A034968(245) = 5 is a divisor of 245, and A034968(246) = 3 is a divisor of 246.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Dahlenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quarterly, Vol. 56, No. 2 (2018), pp. 163-166; alternative link. [Wayback Machine link]

Cf. A007623, A034968, A118363.

sf[n_] := Module[{s = 0, i = 2, k = n}, While[k > 0, k = Floor[n/i!]; s = s + (i - 1)*k; i++]; n - s]; fnQ[n_] := Divisible[n, sf[n]]; aQ[n_] := AllTrue[n + Range[0, 2], fnQ]; Select[Range[400000], aQ] (* after Jean-François Alcover at A034968 *)

A328207 Starts of runs of 4 consecutive factorial base Niven numbers (A118363).

Original entry on oeis.org

9320542, 11397166, 29048470, 29394574, 40469902, 40816006, 58467310, 72657574, 84079006, 101730310, 178911502, 200716054, 283088806, 479329774, 485213542, 499403806, 528476542, 530553166, 544743430, 559625902, 559972006, 574162270, 603235006, 617425270, 641652550

Offset: 1

Amiram Eldar, Oct 07 2019

Dahlenberg & Edgar proved that this sequence is infinite and that there are no consecutive runs of 5 or more factorial base Niven numbers.

a(1)-a(18) were calculated by Dahlenberg & Edgar.

			9320542 is in the sequence since 9320542, 9320543, 9320544 and 9320545 are all in A118363: A034968(9320542) = 22 is a divisor of 9320542, A034968(9320543) = 23 is a divisor of 9320543, A034968(9320544) = 18 is a divisor of 9320544, and A034968(9320545) = 19 is a divisor of 9320545.

Amiram Eldar, Table of n, a(n) for n = 1..512
Paul Dahlenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quarterly, Vol. 56, No. 2 (2018), pp. 163-166; alternative link. [Wayback Machine link]

Cf. A007623, A034968, A118363.

sf[n_] := Module[{s = 0, i = 2, k = n}, While[k > 0, k = Floor[n/i!]; s = s + (i - 1)*k; i++]; n - s]; fnQ[n_] := Divisible[n, sf[n]]; aQ[n_] := AllTrue[n + Range[0, 3], fnQ]; Select[Range[10^8], aQ] (* after Jean-François Alcover at A034968 *)

A377385 Factorial-base Niven numbers (A118363) k such that k/f(k) is also a factorial-base Niven number, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 27, 36, 40, 48, 54, 72, 80, 96, 108, 120, 135, 144, 168, 175, 180, 192, 208, 210, 240, 280, 288, 336, 360, 384, 420, 432, 468, 480, 490, 572, 576, 594, 600, 630, 720, 732, 740, 750, 780, 784, 819, 840, 846, 861, 864, 888, 900, 924, 936, 945, 980, 984

Offset: 1

Amiram Eldar, Oct 27 2024

			8 is a term since 8/f(8) = 4 is an integer and also 4/f(4) = 2 is an integer.

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

Subsequence of A118363.
Subsequences: A000142, A377386.
Cf. A034968, A377384.
Analogous sequences: A376616 (binary), A377209 (Zeckendorf).

fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; q[k_] := Module[{f = fdigsum[k]}, Divisible[k, f] && Divisible[k/f, fdigsum[k/f]]]; Select[Range[1000], q]

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
is(k) = {my(f = fdigsum(k)); !(k % f) && !((k/f) % fdigsum(k/f));}

A377386 Factorial-base Niven numbers (A118363) k such that m = k/f(k) and m/f(m) are also factorial-base Niven numbers, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 36, 40, 48, 54, 72, 80, 96, 108, 120, 135, 144, 180, 192, 240, 280, 288, 360, 384, 432, 480, 576, 594, 600, 720, 840, 864, 1200, 1215, 1225, 1296, 1344, 1440, 1680, 1728, 1800, 2160, 2240, 2352, 2400, 2520, 2592, 2704, 2730, 2880, 3000

Offset: 1

Amiram Eldar, Oct 27 2024

			16 is a term since 16/f(16) = 4 is an integer, 4/f(4) = 2 is an integer, and 2/f(2) = 2 is an integer.

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

Subsequence of A118363 and A377385.
A000142 is a subsequence.
Cf. A034968, A377384.
Analogous sequences: A376617 (binary), A377210 (Zeckendorf).

fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; q[k_] := Module[{f = fdigsum[k], f2, m, n}, IntegerQ[m = k/f] && Divisible[m, f2 = fdigsum[m]] && Divisible[n = m/f2, fdigsum[n]]]; Select[Range[3000], q]

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
is(k) = {my(f = fdigsum(k), f2, m); if(k % f, return(0)); m = k/f; f2 = fdigsum(m); !(m % f2) && !((m/f2) % fdigsum(m/f2)); }

A347495 Factorial base Niven numbers (A118363) with a record gap to the next factorial base Niven number.

Original entry on oeis.org

1, 2, 9, 12, 30, 40, 60, 192, 224, 318, 550, 640, 1136, 1989, 4875, 4980, 23355, 24272, 24378, 40131, 60192, 63872, 80472, 238680, 280140, 2027340, 2872620, 3622068, 13400475, 21293094, 25399080, 28584626, 111020840, 278690360, 355419734, 398884590, 834592590

Offset: 1

Amiram Eldar, Sep 03 2021

The corresponding gaps are 1, 2, 3, 4, 5, 8, 10, 12, 16, 18, 20, 32, 34, 39, 52, 55, 60, 67, 82, 85, 90, 96, 154, 174, 210, 216, 222, 268, 297, 318, 336, 346, 430, 466, 517, 546, 604, ...

			The first 8 factorial base Niven numbers are 1, 2, 4, 6, 8, 9, 12 and 16. The gaps between them are 1, 2, 2, 2, 1, 3 and 4. The record gaps, 1, 2, 3 and 4, occur after the terms 1, 2, 9 and 12.

Cf. A034968, A118363, A328205, A337076, A337077, A347496.

fivenQ[n_] := Module[{s = 0, i = 2, k = n}, While[k > 0, k = Floor[n/i!]; s = s + (i - 1)*k; i++]; Divisible[n, n - s]]; gapmax = 0; n1 = 1; s = {}; Do[If[fivenQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, 10^5}]; s (* after Jean-François Alcover at A034968 *)

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

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[#]] &]

A342426 Niven numbers in base 3/2: numbers divisible by their sum of digits in fractional base 3/2 (A244040).

Original entry on oeis.org

1, 2, 6, 9, 14, 21, 40, 42, 56, 72, 84, 108, 110, 120, 126, 130, 143, 154, 156, 162, 165, 168, 169, 176, 180, 182, 189, 198, 220, 225, 231, 243, 252, 280, 288, 297, 306, 308, 320, 322, 330, 336, 348, 350, 364, 390, 423, 430, 432, 459, 460, 462, 480, 490, 504

Offset: 1

Amiram Eldar, Mar 11 2021

			6 is a term since its representation in base 3/2 is 210 and 2 + 1 + 0 = 3 is a divisor of 6.
9 is a term since its representation in base 3/2 is 2100 and 2 + 1 + 0 + 0 = 3 is a divisor of 9.

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

Cf. A024629, A244040.
Subsequences: A342427, A342428, A342429.
Similar sequences: A005349 (decimal), A049445 (binary), A064150 (ternary), A064438 (quaternary), A064481 (base 5), A118363 (factorial), A328208 (Zeckendorf), A328212 (lazy Fibonacci), A331085 (negaFibonacci), A333426 (primorial), A334308 (base phi), A331728 (negabinary).

s[0] = 0; s[n_] := s[n] = s[2*Floor[n/3]] + Mod[n, 3]; q[n_] := Divisible[n, s[n]]; Select[Range[500], q]

cp's OEIS Frontend

A328205 Numbers m such that m and m+1 are consecutive factorial base Niven numbers (A118363).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A328206 Starts of runs of 3 consecutive factorial base Niven numbers (A118363).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A328207 Starts of runs of 4 consecutive factorial base Niven numbers (A118363).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A377385 Factorial-base Niven numbers (A118363) k such that k/f(k) is also a factorial-base Niven number, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A377386 Factorial-base Niven numbers (A118363) k such that m = k/f(k) and m/f(m) are also factorial-base Niven numbers, where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A347495 Factorial base Niven numbers (A118363) with a record gap to the next factorial base Niven number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

Magma

Magma

Maple

Mathematica

PARI

Python

Sage

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