A002998 -id:A002998 - cp's OEIS frontend

A245062 Array read by upward antidiagonals: Niven (or Harshad) numbers arranged in rows by their digit sums.

1, 2, 10, 3, 20, 100, 4, 12, 110, 1000, 5, 40, 21, 200, 10000, 6, 50, 112, 30, 1010, 100000, 7, 24, 140, 220, 102, 1100, 1000000, 8, 70, 42, 230, 400, 111, 2000, 10000000, 9, 80, 133, 60, 320, 1012, 120, 10010, 100000000, 190, 18, 152, 322, 114, 410, 1120, 201, 10100, 1000000000

Offset: 1

L. Edson Jeffery, Jul 10 2014

The n-th row contains in increasing order all multiples of n with digit sum n.

See A005349 for definitions and references.

			Array begins as:
  1  10  100  1000  10000  100000  1000000  10000000  100000000  1000000000
  2  20  110   200   1010    1100     2000     10010      10100       11000
  3  12   21    30    102     111      120       201        210         300
  4  40  112   220    400    1012     1120      1300       2020        2200
  5  50  140   230    320     410      500      1040       1130        1220
  6  24   42    60    114     132      150       204        222         240
  7  70  133   322    511     700     1015      1141       1204        1330
  8  80  152   224    440     512      800      1016       1160        1232
  9  18   27    36     45      54       63        72         81          90
190 280  370   460    550     640      730       820        910        1090

Alois P. Heinz, Antidiagonals n = 1..70, flattened
Eric W. Weisstein, Harshad Number, from MathWorld.

Cf. A002998 (column 1), A245065 (column 2).
Cf. A011557 (row 1), A069537 (row 2), A052217 (row 3), A063997 (row 4), A069540 (row 5), A062768 (row 6), A063416 (row 7), A069543 (row 8), A052223 (row 9).
Cf. A082260 (main diagonal).
Cf. A007953, A005349 (Niven or Harshad numbers).
Cf. A082259.

A082259 Triangle read by rows in which the n-th row contains n multiples of n with digit sum n.

Original entry on oeis.org

1, 2, 20, 3, 12, 21, 4, 40, 112, 220, 5, 50, 140, 230, 320, 6, 24, 42, 60, 114, 132, 7, 70, 133, 322, 511, 700, 1015, 8, 80, 152, 224, 440, 512, 800, 1016, 9, 18, 27, 36, 45, 54, 63, 72, 81, 190, 280, 370, 460, 550, 640, 730, 820, 910, 1090, 209, 308, 407, 506, 605, 704, 803, 902, 2090, 3080, 4070

Offset: 1

Amarnath Murthy, Apr 12 2003

			Triangle begins:
1
2, 20;
3, 12,  21;
4, 40, 112, 220;
5, 50, 140, 230, 320;
6, 24,  42,  60, 114, 132;
7, 70, 133, 322, 511, 700, 1015;
8, 80, 152, 224, 440, 512,  800, 1016;
9, 18,  27,  36,  45,  54,   63,   72,  81;

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A082260, A082261, A082262, A082263, A002998.

Table[Take[Select[n Range[10000],Total[IntegerDigits[#]]==n&],n],{n,20}] // Flatten (* Harvey P. Dale, Sep 16 2019 *)

Corrected by Anne Donovan, May 29 2003

Corrected and extended by David Wasserman, Aug 26 2004

A082260 a(n) = n-th multiple of n with digit sum n.

Original entry on oeis.org

1, 20, 21, 220, 320, 132, 1015, 1016, 81, 1090, 4070, 516, 2353, 2534, 1185, 3760, 3842, 846, 5662, 14960, 3738, 7546, 12857, 7296, 28825, 25298, 6885, 39088, 43877, 78960, 61969, 78368, 64977, 98872, 258965, 69984, 187849, 367688, 199758

Offset: 1

Amarnath Murthy, Apr 12 2003

Alois P. Heinz, Table of n, a(n) for n = 1..100

Main diagonal of A082259 and of A245062.
Row sums give A082261.
Cf. A082262, A082263, A002998.

Corrected and extended by Ray Chandler, Oct 08 2005

A082262 Average of row n of A082259.

Original entry on oeis.org

1, 11, 12, 94, 149, 63, 394, 404, 45, 604, 1244, 294, 1246, 1391, 705, 2149, 2087, 593, 3466, 8930, 2301, 4927, 7025, 4875, 17809, 15083, 5052, 24850, 26858, 67065, 42979, 56453, 56040, 75400, 152351, 59193, 108631, 228827, 145485, 1366618, 465026

Offset: 1

Amarnath Murthy, Apr 12 2003

Cf. A082259, A082260, A082261, A082263, A002998.

a(n) = my(s=0, k=1); for (i=1, n, while (sumdigits(k*n) != n, k++); s += k*n; k++;); s/n; \\ Michel Marcus, Mar 26 2020

a(n) = A082261(n)/n.

Corrected and extended by Ray Chandler, Oct 08 2005

A131382 Minimal number m such that Sum_digits(n*m)=n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 19, 4, 19, 19, 13, 28, 28, 11, 46, 199, 19, 109, 73, 37, 199, 73, 37, 271, 172, 1333, 289, 559, 1303, 847, 1657, 833, 1027, 1576, 1282, 17497, 4339, 2119, 2323, 10909, 11111, 12826, 14617, 14581, 16102, 199999, 17449, 38269

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 09 2007

			n=23 --> a=73 because 23*73 = 1679 and 1+6+7+9=23.
n=34 --> a=847 because 34*847 = 28798 and 2+8+7+9+8=34.

Peter Lomax, Table of n, a(n) for n = 1..1000 (first 90 terms from T. D. Noe)
H. Fredricksen, E. J. Ionascu, F. Luca, and P. Stanica, Minimal Niven numbers, arXiv:0803.0477 [math.NT], 2008.
Peter Lomax, Phix program for generating 1000 terms

Cf. A005349, A002998.

P:=proc(n) local i,j,k,w; for i from 1 by 1 to n do for j from 1 to n do w:=0; k:=i*j; while k>0 do w:=w+k-(trunc(k/10)*10); k:=trunc(k/10); od; if i=w then print(j); break; fi; od; od; end: P(1000000);

m[n_]:=Module[{m=1},While[Total[IntegerDigits[m*n]]!=n,m++];m]; Array[m,60] (* Harvey P. Dale, Sep 28 2013 *)

a(n)=my(k);while(sumdigits(k+=n)!=n,); k/n \\ Charles R Greathouse IV, Feb 01 2013

a(n) = A002998(n) / n. - Michel Marcus, Dec 10 2012

A003323 Multicolor Ramsey numbers R(3,3,...,3), where there are n 3's.

Original entry on oeis.org

2, 3, 6, 17

Offset: 0

N. J. A. Sloane

Definition: if the edges of a complete graph with at least a(n) nodes are colored with n colors then there is always a monochromatic triangle, and a(n) is the smallest number with this property.
Has it been proved that a(4)=62, or is it just an upper bound? - N. J. A. Sloane, Jun 12 2016
62 is an upper bound. It is probably not the correct value, which is likely closer to the lower bound of 51. - Jeremy F. Alm, Jun 12 2016
From Pontus von Brömssen, Jul 23 2021 (updated Mar 13 2025): (Start)
According to the survey by Radziszowski, the following are the best known bounds:
    51 <= a(4) <=    62,
   162 <= a(5) <=   307,
   538 <= a(6) <=  1838,
  1698 <= a(7) <= 12861.
(End)
In general, if a(n)=r then a(n+1) <= n*(r-1) + r + 1 = (n+1)*(r-1) + 2. - Roderick MacPhee, Mar 03 2023

			a(2)=6 since in a party with at least 6 people, there are three people mutually acquainted or three people mutually unacquainted.

G. Berman and K. D. Fryer, Introduction to Combinatorics. Academic Press, NY, 1972, p. 175.
S. Fettes, R. Kramer, S. Radziszowski, An upper bound of 62 on the classical Ramsey number R(3,3,3,3), Ars Combin. 72 (2004), 41-63.
H. W. Gould, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Shalom Eliahou, An adaptive upper bound on the Ramsey numbers R(3,...,3), Integers 20 (2020), Paper No. A54, 7 pp; arXiv:1912.05353 [math.CO], 2019.
H. W. Gould, Letters to N. J. A. Sloane, 1974
R. E. Greenwood and A. M. Gleason, Combinatorial relations and chromatic graphs, Canad. J. Math., 7 (1955), 1-7.
Stanisław Radziszowski, Small Ramsey numbers, The Electronic Journal of Combinatorics, Dynamic Surveys, DS1 (ver. 17, 2024).

Cf. A045652, A343607.

A073591(n) is an upper bound on a(n).

The limit of a(n)^(1/n) exists and is at least 3.28 (possibly infinite). (See the survey by Radziszowski.) - Pontus von Brömssen, Jul 23 2021 (updated Mar 13 2025)
a(n) = min {k >= 0; A343607(k) > n}. - Pontus von Brömssen, Aug 01 2021
For n >= 4, a(n) <= n!*(e-1/6) + 1. - Elijah Beregovsky, Mar 22 2023

Upper bound and additional comments from D. G. Rogers, Aug 27 2006
Better definition from Max Alekseyev, Jan 12 2008
Comment corrected by Brian Kell, Feb 14 2010
Changed a(4) to 62, following Fettes et al. - Jeremy F. Alm, Jun 08 2016
Entry revised by N. J. A. Sloane, Jun 12 2016
a(4) and a(5) deleted (since they are not known), a(0) prepended by Pontus von Brömssen, Aug 01 2021

A069035 Smallest proper multiple of n with digit sum n.

Original entry on oeis.org

10, 20, 12, 40, 50, 24, 70, 80, 18, 190, 209, 48, 247, 266, 195, 448, 476, 198, 874, 3980, 399, 2398, 1679, 888, 4975, 1898, 999, 7588, 4988, 39990, 8959, 17888, 42999, 28798, 57995, 29988, 37999, 59888, 49998, 699880, 177899, 88998, 99889, 479996

Offset: 1

Amarnath Murthy, Apr 02 2002

David A. Corneth, Table of n, a(n) for n = 1..274

Cf. A002998, A007953.

ssm[n_]:=Module[{k=2},While[Total[IntegerDigits[k*n]]!=n,k++];k*n]; Array[ ssm,50] (* Harvey P. Dale, Nov 24 2012 *)

a(n) = {my(m=2*n); while (sumdigits(m) != n, m+=n); m;} \\ Michel Marcus, Feb 24 2016

Offset corrected by Harvey P. Dale, Nov 24 2012

A082261 Row sums in A082259.

Original entry on oeis.org

1, 22, 36, 376, 745, 378, 2758, 3232, 405, 6040, 13684, 3528, 16198, 19474, 10575, 34384, 35479, 10674, 65854, 178600, 48321, 108394, 161575, 117000, 445225, 392158, 136404, 695800, 778882, 2011950, 1332349, 1806496, 1849320, 2563600

Offset: 1

Amarnath Murthy, Apr 12 2003

Cf. A082259, A082260, A082262, A082263, A002998.

Do[s = t = 0; i = 1; While[t < n, If[Plus @@ IntegerDigits[i*n] == n, t++; s += i*n]; i++ ]; Print[s], {n, 1, 50}] (* Ryan Propper, Jul 16 2005 *)

Corrected and extended by Ryan Propper, Jul 16 2005

A082263 Triangle A082259 with row n divided by n.

Original entry on oeis.org

1, 1, 10, 1, 4, 7, 1, 10, 28, 55, 1, 10, 28, 46, 64, 1, 4, 7, 10, 19, 22, 1, 10, 19, 46, 73, 100, 145, 1, 10, 19, 28, 55, 64, 100, 127, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 28, 37, 46, 55, 64, 73, 82, 91, 109, 19, 28, 37, 46, 55, 64, 73, 82, 190, 280, 370, 4, 7, 13, 16, 19, 22, 28, 31, 34, 37

Offset: 1

Amarnath Murthy, Apr 12 2003

			Triangle begins:
1
1 10
1 4 7
1 10 28 55
1 10 28 46 64
1 4 7 10 19 22

Cf. A082259, A082260, A082261, A082262, A002998.

Corrected and extended by Ray Chandler, Oct 08 2005

A003322 Necklace permutations.

Original entry on oeis.org

1, 1, 1, 2, 4, 14, 57, 347, 2375, 20752, 200805, 2293192, 28097136, 381963696, 5651257984, 87997162749, 1478237810536, 26259254617337, 491527066667949, 9640606422165480

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. W. Gould, Letters to N. J. A. Sloane, 1974.
D. E. Knuth, Computer science and its relation to mathematics, Amer. Math. Monthly, 81 (1974), 323-343. doi: 10.2307/2318994.
D. E. Knuth, H. Wilf, C. L. Mallows, & D. Klarner, Correspondence, 1994,
Index entries for sequences related to necklaces.

Corrected and extended by Don Knuth, Nov 15 1995

cp's OEIS Frontend

A245062 Array read by upward antidiagonals: Niven (or Harshad) numbers arranged in rows by their digit sums.

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A082259 Triangle read by rows in which the n-th row contains n multiples of n with digit sum n.

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A082260 a(n) = n-th multiple of n with digit sum n.

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A082262 Average of row n of A082259.

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A131382 Minimal number m such that Sum_digits(n*m)=n.

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Maple

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A003323 Multicolor Ramsey numbers R(3,3,...,3), where there are n 3's.

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A069035 Smallest proper multiple of n with digit sum n.

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Mathematica

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A082261 Row sums in A082259.

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A082263 Triangle A082259 with row n divided by n.

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