A069540 -id:A069540 - cp's OEIS frontend

A088407 a(n) = A069540(n)/5.

1, 10, 28, 46, 64, 82, 100, 208, 226, 244, 262, 280, 406, 424, 442, 460, 604, 622, 640, 802, 820, 1000, 2008, 2026, 2044, 2062, 2080, 2206, 2224, 2242, 2260, 2404, 2422, 2440, 2602, 2620, 2800, 4006, 4024, 4042, 4060, 4204, 4222, 4240, 4402, 4420, 4600

Offset: 1

Ray Chandler, Sep 29 2003

nonn

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

Cf. A069540.

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.

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

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
```

A279768 Numbers n such that the sum of digits of 8n equals 16.

Original entry on oeis.org

11, 47, 56, 74, 83, 92, 101, 110, 119, 137, 146, 173, 182, 191, 209, 218, 227, 245, 272, 281, 299, 308, 317, 326, 335, 344, 353, 398, 407, 416, 434, 443, 452, 470, 479, 488, 506, 524, 533, 542, 551, 560, 569, 578, 605, 614, 632, 641, 659, 668, 677, 695

Offset: 1

M. F. Hasler, Dec 23 2016

Inspired by A088410 = A069543/8 and A279769 (the analog for 9).

Cf. A007953 (digital sum), 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@ 700, Total@ IntegerDigits[8 #] == 16 &] (* Michael De Vlieger, Dec 23 2016 *)

PARI
```
is(n)=sumdigits(8*n)==16
```

A279775 Numbers k such that the sum of digits of 5k equals 10.

Original entry on oeis.org

11, 29, 38, 47, 56, 65, 74, 83, 92, 101, 110, 128, 146, 164, 182, 209, 218, 227, 236, 245, 254, 263, 272, 281, 290, 308, 326, 344, 362, 380, 407, 416, 425, 434, 443, 452, 461, 470, 488, 506, 524, 542, 560, 605, 614, 623, 632, 641, 650, 668, 686, 704, 722, 740, 803, 812, 821, 830, 848, 866, 884, 902, 920

Offset: 1

M. F. Hasler, Dec 23 2016

Inspired by A088407 = A069540/5 and A279769 (the analog for 9).

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A007953 (digital sum), A279772 (sumdigits(2n) = 4), A279773 (sumdigits(3n) = 6), A279774 (sumdigits(4n) = 8), 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@ 920, Total@ IntegerDigits[5 #] == 10 &] (* Michael De Vlieger, Dec 23 2016 *)

select( is(n)=sumdigits(5*n)==10, [0..999])

def ok(n): return sum(map(int, str(5*n))) == 10
print([k for k in range(921) if ok(k)]) # Michael S. Branicky, Nov 29 2021

A279770 Numbers n such that the sum of digits of 7n equals 14.

Original entry on oeis.org

11, 38, 47, 56, 65, 74, 83, 92, 101, 110, 119, 155, 164, 182, 191, 209, 218, 236, 245, 263, 272, 299, 308, 317, 326, 335, 344, 353, 362, 380, 389, 416, 434, 452, 461, 470, 479, 488, 506, 515, 533, 560, 578, 587, 596, 605, 623, 632, 650, 659, 686, 722, 731

Offset: 1

M. F. Hasler, Dec 23 2016

Inspired by A088409 = A063416/7 and A279769 (the analog for 9).

Cf. A007953 (digital sum), A279772 (sumdigits(2n) = 4), A279773 (sumdigits(3n) = 6), A279774 (sumdigits(4n) = 8), A279775 (sumdigits(5n) = 10), A279776 (sumdigits(6n) = 12), 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@ 731, Total@ IntegerDigits[7 #] == 14 &] (* Michael De Vlieger, Dec 23 2016 *)

PARI
```
is(n)=sumdigits(7*n)==14
```

cp's OEIS Frontend

A088407 a(n) = A069540(n)/5.

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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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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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A069543 Multiples of 8 with digit sum 8.

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

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

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A279768 Numbers n such that the sum of digits of 8n equals 16.

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A279775 Numbers k such that the sum of digits of 5k equals 10.

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