A001101 -id:A001101 - cp's OEIS frontend

A334371 Starts of runs of 3 consecutive Moran numbers (A001101).

3031, 13116, 46824, 201614, 456325, 1310412, 1499434, 1825225, 2217620, 2318423, 2522540, 2784634, 3132380, 3276024, 3931226, 4013113, 4555476, 5017340, 5211380, 6309602, 6338910, 6526835, 7197154, 8678920, 9108023, 9258002, 10256420, 10533620, 10614266, 10810824

Offset: 1

Amiram Eldar, Apr 25 2020

			3031 is a term since 3031/(3+0+3+1) = 433, 3032/(3+0+3+2) = 379 and 3033/(3+0+3+3) = 337 are all primes.

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

Subsequence of A001101, A085775 and A154701.

Cf. A235397, A334346.

moranQ[n_] := PrimeQ[n / Plus @@ IntegerDigits[n]]; m = moranQ /@ Range[3]; seq = {}; Do[If[And @@ m, AppendTo[seq, k - 3]]; m = Join[Rest[m], {moranQ[k]}], {k, 4, 10^6}]; seq

A334372 Starts of runs of 4 consecutive Moran numbers (A001101).

Original entry on oeis.org

21481224, 22314620, 25502420, 25502421, 32432425, 130062260, 147026913, 713021425, 922216713, 938710112, 1012101135, 1019292153, 1113068913, 1420791155, 1545743565, 1671500190, 1805406154, 1941702882, 2010317425, 2027025025, 2200277555, 2307662313, 2437253313

Offset: 1

Amiram Eldar, Apr 25 2020

			21481224 is a term since 21481224/(2+1+4+8+1+2+2+4) = 895051, 21481225/(2+1+4+8+1+2+2+5) = 859249, 21481226/(2+1+4+8+1+2+2+6) = 826201 and 21481227/(2+1+4+8+1+2+2+7) = 795601 are all primes.

Amiram Eldar, Table of n, a(n) for n = 1..197 (terms below 10^11)

Subsequence of A001101, A085775, A141769 and A334371.

Cf. A235397.

moranQ[n_] := PrimeQ[n / Plus @@ IntegerDigits[n]]; m = moranQ /@ Range[4]; seq = {}; Do[If[And @@ m, AppendTo[seq, k - 4]]; m = Join[Rest[m], {moranQ[k]}], {k, 5, 3 * 10^7}]; seq

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)).

A334373 Starts of runs of 5 consecutive Moran numbers (A001101).

Original entry on oeis.org

25502420, 5301223225, 13242121221, 32005512020, 74761736450, 213415171233, 221221400424, 232212103220, 243857053493, 685392911334, 732258727252, 889011113804, 905191111482, 1013460525033, 1080719141080, 1229198438214, 1461057000513, 1961972092132, 2157993351414

Offset: 1

Amiram Eldar, Apr 25 2020

			25502420 is a term since 25502420, 25502421, 25502422, 25502423 and 25502424 are all Moran numbers.

Giovanni Resta, Table of n, a(n) for n = 1..230 (terms < 10^14)

Subsequence of A001101, A085775, A330928, A334371 and A334372.

Cf. A235397.

moranQ[n_] := PrimeQ[n / Plus @@ IntegerDigits[n]]; m = moranQ /@ Range[5]; seq = {}; Do[If[And @@ m, AppendTo[seq, k - 5]]; m = Join[Rest[m], {moranQ[k]}], {k, 6, 10^8}]; seq

Terms a(6) and beyond from Giovanni Resta, Apr 27 2020

A337861 Numbers that can be written as the sum of two Moran numbers (see A001101).

Original entry on oeis.org

36, 39, 42, 45, 48, 54, 60, 63, 66, 69, 72, 81, 84, 87, 90, 102, 105, 108, 111, 126, 129, 132, 135, 138, 141, 144, 147, 151, 153, 154, 156, 159, 160, 162, 168, 170, 171, 173, 174, 175, 177, 178, 179, 180, 183, 189, 192, 194, 195, 196, 197, 198, 201, 208, 211

Offset: 1

Marius A. Burtea, Oct 21 2020

			36 = 18 + 18 = A001101(1) + A001101(1), so 36 is a term.
39 = 18 + 21 = A001101(1) + A001101(2), so 39 is a term.
87 = 42 + 45 = A001101(4) + A001101(5), so 87 is a term.

Cf. A001101, A005349, A337853.

moran:=func; [n:n in [1..220] | #RestrictedPartitions(n,2,{k:k in [1..n-1] | moran(k)}) ne 0];

m = 211; morans = Select[Range[m], PrimeQ[#/Plus @@ IntegerDigits[#]] &]; Select[Range[m], Length[IntegerPartitions[#, {2}, morans]] > 0 &] (* Amiram Eldar, Oct 21 2020 *)

A334376 a(n) is the sum of digits of the n-th Moran number (A001101).

Original entry on oeis.org

9, 3, 9, 6, 9, 9, 12, 3, 6, 9, 7, 8, 9, 12, 9, 10, 15, 18, 3, 9, 11, 6, 12, 13, 9, 14, 15, 9, 10, 12, 21, 6, 11, 9, 12, 15, 13, 7, 12, 14, 9, 15, 18, 16, 9, 17, 15, 18, 9, 10, 12, 18, 18, 21, 9, 11, 12, 18, 24, 15, 18, 27, 3, 8, 9, 10, 15, 18, 3, 7, 9, 12, 16

Offset: 1

Rémy Sigrist, Apr 25 2020

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

Cf. A001101, A007953, A334375.

for (n=1, 1168, if (n%(s=sumdigits(n))==0 && isprime(n/s), print1 (s", ")))

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

A349485 Moran numbers whose arithmetic derivative is also a Moran number (A001101).

Original entry on oeis.org

18, 27, 153, 803, 1101, 1503, 1926, 3070, 3077, 3546, 4577, 6246, 6315, 8717, 10566, 11646, 14093, 15310, 15426, 18456, 24936, 30617, 33576, 34326, 43079, 50418, 59026, 62004, 69781, 71009, 71802, 72587, 74616, 77593, 80118, 94056, 110138, 111546, 112626, 113166

Offset: 1

Marius A. Burtea, Nov 20 2021

Conjecture: The sequence is infinite.

			18 = A001101(1) and 18' = 21 = A001101(2), so 18 is a term.
153 = A001101(13) and 153' = 111 = A001101(8), so 153 is a term.

Cf. A001101 (Moran number), A003415 (arithmetic derivative).

f:=func; moran:=func; [n:n in [2..114000]| moran(n) and moran(Floor(f(n)))];

moranQ[n_] := PrimeQ[n / Plus @@ IntegerDigits[n]]; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[120000], And @@ moranQ /@ {#, d[#]} &] (* Amiram Eldar, Nov 20 2021 *)

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

A085775 Numbers k such that (k / sum of digits of k) and (k+1 / sum of digits of k+1) are both prime.

Original entry on oeis.org

152, 803, 1016, 1853, 3031, 3032, 3438, 7361, 7542, 7587, 8226, 8337, 10095, 10278, 10307, 11354, 11646, 13116, 13117, 13881, 17153, 21434, 21906, 23412, 26221, 28824, 30254, 31112, 32166, 34218, 35513, 38322, 40335, 41058, 44373, 45380

Offset: 1

Jason Earls, Jul 23 2003

			152 is a term since 152/(1+5+2) = 19 and 153/(1+5+3) = 17 are both prime.

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

Subsequence of A001101 and A330927.

moranQ[n_] := PrimeQ[n / Plus @@ IntegerDigits[n]]; Select[Range[50000], moranQ[#] && moranQ[#+1] &] (* Amiram Eldar, Apr 25 2020 *)

Offset corrected by Amiram Eldar, Apr 25 2020

A235397 The first term of the least sequence of n consecutive Moran numbers.

Original entry on oeis.org

18, 152, 3031, 21481224, 25502420, 4007565001480, 2196125475223740, 905295493763807066010

Offset: 1

Carlos Rivera, Jan 09 2014

A number n is a Moran number if n divided by the sum of its decimal digits is prime.
From Amiram Eldar, Apr 25 2020: (Start)
Jens Kruse Andersen found that a(7) <= 2196125475223740 and a(8) <= 905295493763807066010 (see Rivera link).
Since Moran numbers (A001101) are also Niven numbers (A005349), this sequence is finite with no more than 20 terms (see A060159). (End)
a(9) <= 270140199032572375590810. - Giovanni Resta, Apr 30 2020

			a(6) = 4007565001480 because
4007565001480 = 40 * 100189125037,
4007565001481 = 41 * 97745487841,
4007565001482 = 42 * 95418214321,
4007565001483 = 43 * 93199186081,
4007565001484 = 44 * 91081022761,
4007565001485 = 45 * 89057000033.

Giovanni Resta, Moran numbers
Carlos Rivera, Puzzle 728. Consecutive Moran numbers, The Prime Puzzles & Problems Connection.

Cf. A001101, A060159, A085775.

isA001101(n)=(k->denominator(k)==1&&isprime(k))(n/sumdigits(n))
a(n)=my(k=n); while(1, forstep(i=k,k-n+1,-1, if(!isA001101(i), k=i+n; next(2))); return(k-n+1)) \\ Charles R Greathouse IV, Jan 10 2014

a(7)-a(8) from Giovanni Resta, Apr 27 2020

cp's OEIS Frontend

A334371 Starts of runs of 3 consecutive Moran numbers (A001101).

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A334372 Starts of runs of 4 consecutive Moran numbers (A001101).

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

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A334373 Starts of runs of 5 consecutive Moran numbers (A001101).

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Extensions

A337861 Numbers that can be written as the sum of two Moran numbers (see A001101).

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Magma

Mathematica

A334376 a(n) is the sum of digits of the n-th Moran number (A001101).

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PARI

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A349485 Moran numbers whose arithmetic derivative is also a Moran number (A001101).

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Magma

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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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