A052216 -id:A052216 - cp's OEIS frontend

A038444 Sums of 2 distinct powers of 10.

11, 101, 110, 1001, 1010, 1100, 10001, 10010, 10100, 11000, 100001, 100010, 100100, 101000, 110000, 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 10000001, 10000010, 10000100, 10001000, 10010000, 10100000, 11000000, 100000001, 100000010, 100000100

Offset: 1

Olivier Gérard

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequence of A052216 and of A007088.

Cf. A018900.

a038444 n = a038444_list !! (n-1)
a038444_list = 11 : f [11] 90 where
   f xs@(x:_) z = ys ++ f ys (10 * z) where
                  ys = (x + z) : map (* 10) xs
-- Reinhard Zumkeller, Jan 28 2015

seq(seq(10^d + 10^j, j=0..d-1), d=1..10); # Robert Israel, Oct 14 2016

Sort[Total/@Subsets[10^Range[0,7],{2}]] (* Harvey P. Dale, Apr 20 2012 *)

a(n)= 10^(n-1-binomial(sqrtint(n*8)\/2, 2)) + 10^((sqrtint(n*8)+1)\2); \\ Ruud H.G. van Tol, Nov 29 2024

from math import isqrt
def A038444(n): return 10**(m:=isqrt(n<<3)+1>>1)+10**(n-1-(m*(m-1)>>1)) # Chai Wah Wu, Mar 11 2025

G.f.: (10*x - 55*x^2 + Sum_{d>=1} (4*10^d+5)*x^((d^2-d)/2+1) - Sum_{d>=1} (445*10^(d-1)+5)*x^((d^2-d)/2+2))/(5*(1-x)*(1-10*x)). - Robert Israel, Oct 14 2016

Offset corrected by Reinhard Zumkeller, Jan 28 2015

A242614 Triangle read by rows: row n contains numbers with sum of digits = n, and not greater than the n-th repunit (cf. A007953 and A002275).

Original entry on oeis.org

0, 1, 2, 11, 3, 12, 21, 30, 102, 111, 4, 13, 22, 31, 40, 103, 112, 121, 130, 202, 211, 220, 301, 310, 400, 1003, 1012, 1021, 1030, 1102, 1111, 5, 14, 23, 32, 41, 50, 104, 113, 122, 131, 140, 203, 212, 221, 230, 302, 311, 320, 401, 410, 500, 1004, 1013, 1022

Offset: 0

Reinhard Zumkeller, Jul 16 2014

Number of terms in row n = A242622(n);
T(n,1) = A051885(n);
T(n,A242622(n)) = A002275(n);
for n > 0: number of repdigit terms in row n = A242627(n).

			The triangle begins:
. 0:  0
. 1:  1
. 2:  2,11
. 3:  3,12,21,30,102,111
. 4:  4,13,22,31,40,103,112,121,130,202, . . . ,1021,1030,1102,1111
. 5:  5,14,23,32,41,50,104,113,122,131, . . . ,11021,11030,11102,11111 .

Reinhard Zumkeller, Rows n = 0..8 of table, flattened

Cf. A007953 (sum of digits), A002275 (repunits).

Cf. A011557, A052216, A052217, A052218, A052219, A052220, A052221, A052222, A052223, A052224, A166311, A235151, A143164, A235225, A235226, A235227, A166370, A235228, A166459, A235229.

a242614 n k = a242614_row n !! (k-1)
a242614_row n = filter ((== n) . a007953) [n .. a002275 n]
a242614_tabf = map a242614_row [0..]

Join[{0},Flatten[Table[Select[Range[FromDigits[PadRight[{},n,1]]], Total[ IntegerDigits[ #]] == n&],{n,5}]]] (* Harvey P. Dale, Oct 08 2019 *)

A055235 Sums of two powers of 3.

Original entry on oeis.org

2, 4, 6, 10, 12, 18, 28, 30, 36, 54, 82, 84, 90, 108, 162, 244, 246, 252, 270, 324, 486, 730, 732, 738, 756, 810, 972, 1458, 2188, 2190, 2196, 2214, 2268, 2430, 2916, 4374, 6562, 6564, 6570, 6588, 6642, 6804, 7290, 8748, 13122, 19684, 19686, 19692, 19710

Offset: 0

Henry Bottomley, Jun 22 2000

T. D. Noe, Rows n = 0..100 of triangle, flattened

Cf. A002262, A003056, A052216.

Partial sums of A135293.

mx = 10; Sort[Flatten[Table[3^x + 3^y, {y, 0, mx}, {x, 0, y}]]] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)
f[n_] := Block[{t = Floor[(Sqrt[1 + 8 (n - 1)] - 1)/2]}, 3^(n - 1 - t*(t + 1)/2) + 3^t]; Array[f, 49] (* Robert G. Wilson v, Oct 08 2011 *)
Total/@Tuples[3^Range[0,10],2]//Union (* Harvey P. Dale, Aug 28 2025 *)

for( n=0,5, for(k=0,n, print1(3^n+3^k",")))

A055235(n)={ my( t=(sqrtint(8*n-7)-1)\2); 3^t+3^(n-1-t*(t+1)/2) }  \\ M. F. Hasler, Oct 08 2011

from math import isqrt
def A055235(n): return 3**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+3**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 08 2025

a(n+1) = 3^(n-trinv(n)*(trinv(n)+1)/2)+3^trinv(n), where trinv(n) = floor((sqrt(1+8*n)-1)/2) = A003056(n) and n-trinv(n)*(trinv(n)+1)/2 = A002262(n). [corrected by M. F. Hasler, Oct 08 2011]

Regarded as a triangle, T(n, k) = 3^n + 3^k, because 3^n + 3^n < 3^(n+1) + 3^0 for all n > 0.

A237424 Numbers of the form (10^a + 10^b + 1)/3.

Original entry on oeis.org

1, 4, 7, 34, 37, 67, 334, 337, 367, 667, 3334, 3337, 3367, 3667, 6667, 33334, 33337, 33367, 33667, 36667, 66667, 333334, 333337, 333367, 333667, 336667, 366667, 666667, 3333334, 3333337, 3333367, 3333667, 3336667

Offset: 1

Ahmad J. Masad, Feb 07 2014

nonn

Has the property that the product of any two (not necessarily distinct) terms has digits in nondecreasing order.
Conjecture: This sequence is in a sense the maximally dense sequence with this nondecreasing products property. That is, it appears that every maximal sequence is either (i) A237424, (ii) a finite set of "extra" terms plus A237424 restricted to b=0 (which is A093137), or (iii) a finite set of "extra" terms plus A237424 restricted to a=b (which is A067275). (There might be one more case, not yet identified.) - David Applegate, Feb 12 2014
See A254143 and link for products a(i)*a(j) in natural order. - Reinhard Zumkeller, Jan 28 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1035 terms from Robert G. Wilson v)
Reinhard Zumkeller, First 10000 products of any two terms of A237424

Cf. A028819, A028820, A028821, A052216, A067275, A093137, A234841, A254143.

a237424 = flip div 3 . (+ 1) . a052216
-- Reinhard Zumkeller, Jan 28 2015

A052216:=[10^(n-1) + 10^(k-1): k in [1..n], n in [1..100]];
A237424:= func< n | (A052216[n]+1)/3 >;
[A237424(n): n in [1..100]]; // G. C. Greubel, Feb 22 2024

Union@ Flatten@ Table[(10^a + 10^b + 1)/3, {a, 0, 8}, {b, a, 8}] (* Robert G. Wilson v, Jan 26 2015 *)
(10^#[[1]]+10^#[[2]]+1)/3&/@Tuples[Range[0,8],2]//Union (* Harvey P. Dale, May 28 2019 *)

list(lim)=my(v=List(),a,t); while(1, for(b=0,a, t=(10^a+10^b+1)/3; if(t>lim, return(Set(v))); listput(v, t)); a++) \\ Charles R Greathouse IV, May 13 2015

from math import isqrt
def A237424(n): return (10**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+10**(n-1-(a*(a+1)>>1))+1)//3 # Chai Wah Wu, Apr 08 2025

A052216=flatten([[10^(n-1) + 10^(k-1) for k in range(1,n+1)] for n in range(1,101)])
def A237424(n): return (A052216[n-1]+1)//3
[A237424(n) for n in range(1,101)] # G. C. Greubel, Feb 22 2024

a(n) = (A052216(n) + 1)/3. - Reinhard Zumkeller, Jan 28 2015

Edited by David Applegate, Feb 07 2014

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
```

A279772 Numbers n such that the sum of digits of 2n equals 4.

Original entry on oeis.org

2, 11, 20, 56, 65, 101, 110, 155, 200, 506, 515, 551, 560, 605, 650, 1001, 1010, 1055, 1100, 1505, 1550, 2000, 5006, 5015, 5051, 5060, 5105, 5150, 5501, 5510, 5555, 5600, 6005, 6050, 6500, 10001, 10010, 10055, 10100, 10505, 10550, 11000, 15005, 15050, 15500

Offset: 1

M. F. Hasler, Dec 23 2016

Inspired by A088404 = A069537/2 and A279769 (the analog for 9).

Cf. A007953 (digital sum), A052216 (sumdigits(n) = 2), 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@ 15500, Total@ IntegerDigits[2 #] == 4 &] (* Michael De Vlieger, Dec 23 2016 *)

select( is(n)=sumdigits(2*n)==4, [1..9999])

cp's OEIS Frontend

A038444 Sums of 2 distinct powers of 10.

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A242614 Triangle read by rows: row n contains numbers with sum of digits = n, and not greater than the n-th repunit (cf. A007953 and A002275).

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A055235 Sums of two powers of 3.

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A237424 Numbers of the form (10^a + 10^b + 1)/3.

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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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