A069537 -id:A069537 - cp's OEIS frontend

A088404 a(n) = A069537(n)/2.

1, 10, 55, 100, 505, 550, 1000, 5005, 5050, 5500, 10000, 50005, 50050, 50500, 55000, 100000, 500005, 500050, 500500, 505000, 550000, 1000000, 5000005, 5000050, 5000500, 5005000, 5050000, 5500000, 10000000, 50000005, 50000050, 50000500

Offset: 1

Ray Chandler, Sep 29 2003

Sequence comprises all integers that are the arithmetic mean of two powers of ten, A011557. - Lewis Chen, Jul 17 2025

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A069537.

A052217 Numbers whose sum of digits is 3.

Original entry on oeis.org

3, 12, 21, 30, 102, 111, 120, 201, 210, 300, 1002, 1011, 1020, 1101, 1110, 1200, 2001, 2010, 2100, 3000, 10002, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000, 20001, 20010, 20100, 21000, 30000, 100002, 100011, 100020, 100101

Offset: 1

Henry Bottomley, Feb 01 2000

From Joshua S.M. Weiner, Oct 19 2012: (Start)
Sequence is a representation of the "energy states" of "multiplex" notation of 3 quantum of objects in a juggling pattern.
0 = an empty site, or empty hand. 1 = one object resides in the site. 2 = two objects reside in the site. 3 = three objects reside in the site. (See A038447.) (End)
A007953(a(n)) = 3; number of repdigits = #{3,111} = A242627(3) = 2. - Reinhard Zumkeller, Jul 17 2014
Can be seen as a table whose n-th row holds the n-digit terms {10^(n-1) + 10^m + 10^k, 0 <= k <= m < n}, n >= 1. Row lengths are then (1, 3, 6, 10, ...) = n*(n+1)/2 = A000217(n). The first and the n last terms of row n are 10^(n-1) + 2 resp. 2*10^(n-1) + 10^k, 0 <= k < n. - M. F. Hasler, Feb 19 2020

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..84 from Vincenzo Librandi, terms 85..1140 from T. D. Noe)

Cf. A069521 to A069530, A069532 to A069537.
Cf. A007953, A218043 (subsequence).
Row n=3 of A245062.
Other digit sums: A011557 (1), A052216 (2), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Other bases: A014311 (binary), A226636 (ternary), A179243 (Zeckendorf).
Cf. A242614, A242627.
Cf. A003056, A002262 (triangular coordinates), A056556, A056557, A056558 (tetrahedral coordinates).

a052217 n = a052217_list !! (n-1)
a052217_list = filter ((== 3) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..100101] | &+Intseq(n) eq 3 ]; // Vincenzo Librandi, Mar 07 2013

Union[FromDigits/@Select[Flatten[Table[Tuples[Range[0,3],n],{n,6}],1],Total[#]==3&]] (* Harvey P. Dale, Oct 20 2012 *)
Select[Range[10^6], Total[IntegerDigits[#]] == 3 &] (* Vincenzo Librandi, Mar 07 2013 *)
Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 18]], {s, IntegerPartitions[3]}]]] (* T. D. Noe, Mar 08 2013 *)

isok(n) = sumdigits(n) == 3; \\ Michel Marcus, Dec 28 2015

apply( {A052217_row(n,s,t=-1)=vector(n*(n+1)\2,k,t++>s&&t=!s++;10^(n-1)+10^s+10^t)}, [1..5]) \\ M. F. Hasler, Feb 19 2020

from itertools import count, islice
def agen(): yield from (10**i + 10**j + 10**k for i in count(0) for j in range(i+1) for k in range(j+1))
print(list(islice(agen(), 40))) # Michael S. Branicky, May 14 2022

from math import comb, isqrt
from sympy import integer_nthroot
def A052217(n): return 10**((m:=integer_nthroot(6*n,3)[0])-(a:=n<=comb(m+2,3)))+10**((k:=isqrt(b:=(c:=n-comb(m-a+2,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1))+10**(c-1-comb(k+(b>k*(k+1)),2)) # Chai Wah Wu, Dec 11 2024

T(n,k) = 10^(n-1) + 10^A003056(k) + 10^A002262(k) when read as a table with row lengths n*(n+1)/2, n >= 1, 0 <= k < n*(n+1)/2. - M. F. Hasler, Feb 19 2020

a(n) = 10^A056556(n-1) + 10^A056557(n-1) + 10^A056558(n-1). - Kevin Ryde, Apr 17 2021

Offset changed from 0 to 1 by Vincenzo Librandi, Mar 07 2013

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

Original entry on oeis.org

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.

A063997 Multiples of 4 whose digits add to 4.

Original entry on oeis.org

4, 40, 112, 220, 400, 1012, 1120, 1300, 2020, 2200, 3100, 4000, 10012, 10120, 10300, 11020, 11200, 12100, 13000, 20020, 20200, 21100, 22000, 30100, 31000, 40000, 100012, 100120, 100300, 101020, 101200, 102100, 103000, 110020, 110200, 111100

Offset: 1

Lisa O. Coulter (lcoulter(AT)stetson.edu), Sep 06 2001

			4 is an element of the sequence, since 4 is a multiple of 4 the sum of whose digits is 4; 220 is an element of the sequence, since 220 = 4*55 and 2 + 2+ 0 = 4.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 218 terms from Harry J. Smith)

Cf. A069521 to A069530, A069532, A069533, A069534, A069535, A069536, A069537, A052217.

Row n=4 of A245062.

Select[4Range[120000],Total[IntegerDigits[#]]==4&] (* Harvey P. Dale, May 07 2011 *)

SumDE(x,y)= { local(s); s=0; while (x>9 && sHarry J. Smith, Sep 05 2009

More terms from Ray Chandler, Sep 28 2003

A062768 Multiples of 6 such that the sum of the digits is equal to 6.

Original entry on oeis.org

6, 24, 42, 60, 114, 132, 150, 204, 222, 240, 312, 330, 402, 420, 510, 600, 1014, 1032, 1050, 1104, 1122, 1140, 1212, 1230, 1302, 1320, 1410, 1500, 2004, 2022, 2040, 2112, 2130, 2202, 2220, 2310, 2400, 3012, 3030, 3102, 3120, 3210, 3300, 4002, 4020, 4110

Offset: 1

Lisa O Coulter (lisa_coulter(AT)my-deja.com), Jul 17 2001

Even numbers with sum of digits equal to 6 are Harshad numbers (A005349). - Davide Rotondo, Sep 04 2020

			60 is a member of the sequence since 60 / 6 = 10 and 6 + 0 = 6; 114 is also an element since 114 is divisible by 6 and 1 + 1+ 4 = 6.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A063416, A069521 to A069530, A069532, A069533, A069534, A069535, A069536, A069537, A052217, A063997, A069540.
Subsequence of A005349.
Row n=6 of A245062.

: var stk: stack; end; minarg := 0; maxarg := 900; n := 6; for k := minarg to maxarg do m := k*n; s := itoa(m); for j := 0 to length(s) - 1 do stack_push(stk,atoi(s[j..j])); end; if sum(stack2array(stk)) = n then write(m," "); end; end;.

Select[ Range[ 6, 4200, 6 ], Plus @@ IntegerDigits[ # ] == 6 & ]

More terms from Klaus Brockhaus, Jul 20 2001

A063416 Multiples of 7 whose sum of digits is equal to 7.

Original entry on oeis.org

7, 70, 133, 322, 511, 700, 1015, 1141, 1204, 1330, 2023, 2212, 2401, 3031, 3220, 4102, 5110, 7000, 10024, 10150, 10213, 10402, 11032, 11221, 11410, 12040, 12103, 13111, 13300, 15001, 20041, 20104, 20230, 21112, 21301, 22120, 23002, 24010

Offset: 1

Klaus Brockhaus, Jul 20 2001

Numbers are all 7 mod 63.

			133 = 19*7 and 1+3+3 = 7, so 133 is a term of this sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 700 terms from Harry J. Smith)

Cf. A069521 to A069530, A069532, A069533, A069534, A069535, A069536, A069537, A052217, A063997, A069540, A062768.

Row n=7 of A245062.

: var stk: stack; end; minarg := 0; maxarg := 5000; n := 7; for k := minarg to maxarg do m := k*n; s := itoa(m); for j := 0 to length(s) - 1 do stack_push(stk,atoi(s[j..j])); end; if sum(stack2array(stk)) = n then write(m," "); end; end;.

Select[Range[7, 25000, 7], Plus @@ IntegerDigits[ # ] == 7 &]

forstep(m=0, 70000, 7, if(vecsum(digits(m))==7, print1(m, ", "))) \\ Harry J. Smith, Aug 20 2009

A069540 Multiples of 5 with digit sum 5.

Original entry on oeis.org

5, 50, 140, 230, 320, 410, 500, 1040, 1130, 1220, 1310, 1400, 2030, 2120, 2210, 2300, 3020, 3110, 3200, 4010, 4100, 5000, 10040, 10130, 10220, 10310, 10400, 11030, 11120, 11210, 11300, 12020, 12110, 12200, 13010, 13100, 14000, 20030, 20120

Offset: 1

Amarnath Murthy, Apr 01 2002

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A069521 to A069530, A069532, A069533, A069534, A069535, A069536, A069537, A052217, A063997.

Row n=5 of A245062.

Select[5*Range[5000],Total[IntegerDigits[#]]==5&] (* Harvey P. Dale, Nov 08 2017 *)

Corrected and extended by Ray Chandler, Sep 28 2003

A069543 Multiples of 8 with digit sum 8.

Original entry on oeis.org

8, 80, 152, 224, 440, 512, 800, 1016, 1160, 1232, 1304, 1520, 2024, 2240, 2312, 2600, 3032, 3104, 3320, 4040, 4112, 4400, 5120, 6200, 8000, 10016, 10160, 10232, 10304, 10520, 11024, 11240, 11312, 11600, 12032, 12104, 12320, 13040, 13112, 13400

Offset: 1

Amarnath Murthy, Apr 01 2002

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A069521 to A069530, A069532, A069533, A069534, A069535, A069536, A069537, A052217, A063997, A069540, A062768, A063416.

Row n=8 of A245062.

More terms from Ray Chandler, Sep 28 2003

A279769 Numbers n such that the sum of digits of 9n is 18.

Original entry on oeis.org

11, 21, 22, 31, 32, 33, 41, 42, 43, 44, 51, 52, 53, 54, 55, 61, 62, 63, 64, 65, 66, 71, 72, 73, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 121, 122, 131, 132, 133, 141

Offset: 1

M. F. Hasler, Dec 18 2016

Differs from A084854 from a(55) = 110 on.
Numbers n such that A008591(n) is a term of A235228. - Felix Fröhlich, Dec 18 2016
The digital sum of 9n is always a multiple of 9, and never zero. For most numbers < 100, the digital sum is equal to 9, but for example in the range [91..110] all numbers except 100 have their digital sum equal to 18. The b-file / graph gives a hint on the "asymptotic" distribution / density of this set. After a "flat" range like that at [91..110] there comes a record gap. Sizes [and upper ends] of record gaps are: 10 [a(2) = 21], 11 [a(56) = 121, a(119) = 231, a(188) = 341, ..., a(553) = 891, a(616) = 1001], 21 [a(671) = 1121], 31 [a(1331) = 2231], ..., 91 [a(4339) = 8891], 101 [a(4621) = 10001], 121 [a(4841) = 11121], 231 [a(9176) = 22231], ..., 891 [a(24217) = 88891], 1001 [a(25213) = 100001], 1121 [a(25928) = 111121], 2231 [a(47510) = 222231], ..., 8891 [a(108577) = 888891], 10001 [a(111574) = 1000001], 11121 [a(113576) = 1111121], 22231 [a(202511) = 2222231], ..., 88891 [a(416215) = 8888891], ... - M. F. Hasler, Dec 22 2016

M. F. Hasler, Table of n, a(n) for n = 1..25212

Cf. A007953 (digital sum), A008591, A084854.
Cf. A279772 (sumdigits(2n) = 4), A279773 (sumdigits(3n) = 6), A279774 (sumdigits(4n) = 8), A279775 (sumdigits(5n) = 10), A279776 (sumdigits(6n) = 12), A279770 (sumdigits(7n) = 14), A279768 (sumdigits(8n) = 16), A279769 (sumdigits(9n) = 18), A279777 (sumdigits(9n) = 27).
Digital sum of m*n equals m: A088404 = A069537/2, A088405 = A052217/3, A088406 = A063997/4, A088407 = A069540/5, A088408 = A062768/6, A088409 = A063416/7, A088410 = A069543/8.
Cf. A005349 (Niven or Harshad numbers), A245062 (arranged in rows by digit sums).
Numbers with given digital sum: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).

Select[Range@ 141, Total@ IntegerDigits[9 #] == 18 &]

is(n) = sumdigits(9*n)==18 \\ Felix Fröhlich, Dec 18 2016

a(n) = A235228(n)/9.

A279777 Numbers k such that the sum of digits of 9k is 27.

Original entry on oeis.org

111, 211, 221, 222, 311, 321, 322, 331, 332, 333, 411, 421, 422, 431, 432, 433, 441, 442, 443, 444, 511, 521, 522, 531, 532, 533, 541, 542, 543, 544, 551, 552, 553, 554, 555, 611, 621, 622, 631, 632, 633, 641, 642, 643, 644, 651, 652, 653, 654, 655, 661

Offset: 1

M. F. Hasler, Dec 23 2016

The digital sum of 9k is always a multiple of 9. For most numbers below 100 it is actually equal to 9. Numbers such that the digital sum of 9k is 18 are listed in A279769. Only every third term of the present sequence is divisible by 3.

The sequence of record gaps [and upper end of the gap] is: 100 [a(2) = 211], 101 [a(221) = 1211], 111 [a(4841) = 11211], 111 [a(10121) = 22311], 111 [a(15752) = 33411], ..., 111 [a(45133) = 88911], 111 [a(50413) = 100011], 211 [a(55253) = 111211], 311 [a(110000) = 222311], ..., 911 [a(380557) = 888911], 1011 [a(411049) = 1000011], 1211 [a(436976) = 1111211], 2311 [a(840281) = 2222311], ..., 8911 [a(2451241) = 8888911], ...

Cf. A008591, A084854, A003991, A004247, A279769 (sumdigits(9n) = 18).
Digital sum of m*n equals m: A088404 = A069537/2, A088405 = A052217/3, A088406 = A063997/4, A088407 = A069540/5, A088408 = A062768/6, A088409 = A063416/7, A088410 = A069543/8.
Numbers with given digital sum: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A007953 (digital sum), A005349 (Niven or Harshad numbers), A245062 (arranged in rows by digit sums).
Cf. A082259.

Select[Range@ 661, Total@ IntegerDigits[9 #] == 27 &] (* Michael De Vlieger, Dec 23 2016 *)

PARI
```
is(n)=sumdigits(9*n)==27
```

cp's OEIS Frontend

A088404 a(n) = A069537(n)/2.

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A052217 Numbers whose sum of digits is 3.

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

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A063997 Multiples of 4 whose digits add to 4.

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A062768 Multiples of 6 such that the sum of the digits is equal to 6.

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A063416 Multiples of 7 whose sum of digits is equal to 7.

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A069540 Multiples of 5 with digit sum 5.

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A069543 Multiples of 8 with digit sum 8.

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