A235225 -id:A235225 - cp's OEIS frontend

A052216 Sums of two powers of 10.

2, 11, 20, 101, 110, 200, 1001, 1010, 1100, 2000, 10001, 10010, 10100, 11000, 20000, 100001, 100010, 100100, 101000, 110000, 200000, 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000, 10000001, 10000010, 10000100, 10001000, 10010000, 10100000, 11000000, 20000000

Offset: 1

Henry Bottomley, Feb 01 2000

Numbers whose digit sum is 2.
A007953(a(n)) = 2; number of repdigits = #{2,11} = A242627(2) = 2. - Reinhard Zumkeller, Jul 17 2014
By extension, numbers k such that digitsum(k)^2 - 1 is prime. (PROOF: For any number k whose digit sum d > 2, d^2 - 1 = (d+1)*(d-1) and thus is not prime.) - Christian N. K. Anderson, Apr 22 2024

			From _Bruno Berselli_, Mar 07 2013: (Start)
The triangular array starts (see formula):
        2;
       11,      20;
      101,     110,     200;
     1001,    1010,    1100,    2000;
    10001,   10010,   10100,   11000,   20000;
   100001,  100010,  100100,  101000,  110000,  200000;
  1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000;
  ...
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (terms 1..48 from Vincenzo Librandi, terms 49..1036 from T. D. Noe)

Subsequence of A069263 and A107679. A038444 is a subsequence.
Sums of n powers of 10: A011557 (1), 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. A002262, A003056, A007953, A242614, A242627, A237424.

a052216 n = a052216_list !! (n-1)
a052216_list = 2 : f [2] 9 where
   f xs@(x:_) z = ys ++ f ys (10 * z) where
                  ys = (x + z) : map (* 10) xs
-- Reinhard Zumkeller, Jan 28 2015, Jul 17 2014

[n: n in [1..10100000] | &+Intseq(n) eq 2]; // Vincenzo Librandi, Mar 07 2013

/* As a triangular array: */ [[10^n+10^m: m in [0..n]]: n in [0..8]]; // Bruno Berselli, Mar 07 2013

t = 10^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
With[{nn=7},Sort[Join[Table[FromDigits[PadRight[{2},n,0]],{n,nn}], FromDigits/@Flatten[Table[Table[Insert[PadRight[{1},n,0],1,i]],{n,nn},{i,2,n+1}],1]]]] (* Harvey P. Dale, Nov 15 2011 *)
Select[Range[10^9], Total[IntegerDigits[#]] == 2&] (* Vincenzo Librandi, Mar 07 2013 *)
T[n_,k_]:=10^(n-1)+10^(k-1); Table[T[n,k],{n,8},{k,n}]//Flatten (* Stefano Spezia, Nov 03 2023 *)

a(n)=my(d=(sqrtint(8*n)-1)\2,t=n-d*(d+1)/2-1); 10^d + 10^t \\ Charles R Greathouse IV, Dec 19 2016

from itertools import count, islice
def agen(): yield from (10**i + 10**j for i in count(0) for j in range(i+1))
print(list(islice(agen(), 34))) # Michael S. Branicky, May 15 2022

from math import isqrt
def A052216(n): return 10**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+10**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 08 2025

def A052216(n,k): return 10^(n-1) + 10^(k-1)
flatten([[A052216(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Feb 22 2024

T(n,k) = 10^(n-1) + 10^(k-1) with 1 <= k <= n.
a(n) = 3*A237424(n) - 1. - Reinhard Zumkeller, Jan 28 2015
a(n) = 10^A003056(n-1) + 10^A002262(n-1). - Chai Wah Wu, Apr 08 2025

A052224 Numbers whose sum of digits is 10.

Original entry on oeis.org

19, 28, 37, 46, 55, 64, 73, 82, 91, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 208, 217, 226, 235, 244, 253, 262, 271, 280, 307, 316, 325, 334, 343, 352, 361, 370, 406, 415, 424, 433, 442, 451, 460, 505, 514, 523, 532, 541, 550, 604, 613, 622, 631, 640

Offset: 1

Henry Bottomley, Feb 01 2000

Proper subsequence of A017173. - Rick L. Shepherd, Jan 12 2009
Subsequence of A227793. - Michel Marcus, Sep 23 2013
A007953(a(n)) = 10; number of repdigits = #{55,22222,1^10} = A242627(10) = 3. - Reinhard Zumkeller, Jul 17 2014
a(n) = A094677(n) for n = 1..28. - Reinhard Zumkeller, Nov 08 2015
The number of terms having <= m digits is the coefficient of x^10 in sum(i=0,9,x^i)^m = ((1-x^10)/(1-x))^m. - David A. Corneth, Jun 04 2016
In general, the set of numbers with sum of base-b digits equal to b is a subset of { (b-1)*k + 1; k = 2, 3, 4, ... }. - M. F. Hasler, Dec 23 2016

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 3921 terms from T. D. Noe)

Cf. A007953, A017173, A228915, A254524.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A242614, A242627.
Cf. A094677.
Cf. A018900, A187813.
Sum of base-b digits equal b: A226636 (b = 3), A226969 (b = 4), A227062 (b = 5), A227080 (b = 6), A227092 (b = 7), A227095 (b = 8), A227238 (b = 9).

a052224 n = a052224_list !! (n-1)
a052224_list = filter ((== 10) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..1000] | &+Intseq(n) eq 10 ]; // Vincenzo Librandi, Mar 10 2013

sd := proc (n) options operator, arrow: add(convert(n, base, 10)[j], j = 1 .. nops(convert(n, base, 10))) end proc: a := proc (n) if sd(n) = 10 then n else end if end proc: seq(a(n), n = 1 .. 800); # Emeric Deutsch, Jan 16 2009

Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 7]], {s, Rest[IntegerPartitions[10]]}]]] (* T. D. Noe, Mar 08 2013 *)
Select[Range[1000], Total[IntegerDigits[#]] == 10 &] (* Vincenzo Librandi, Mar 10 2013 *)

isok(n) = sumdigits(n) == 10; \\ Michel Marcus, Dec 28 2015

\\ This algorithm needs a modified binomial.
C(n, k)=if(n>=k, binomial(n, k), 0)
\\ ways to roll s-q with q dice having sides 0 through n - 1.
b(s, q, n)=if(s<=q*(n-1), s+=q; sum(i=0, q-1, (-1)^i*C(q, i)*C(s-1-n*i, q-1)), 0)
\\ main algorithm; this program applies to all sequences of the form "Numbers whose sum of digits is m."
a(n,{m=10}) = {my(q); q = 2; while(b(m, q, 10) < n, q++); q--; s = m; os = m; r=0; while(q, if(b(s, q, 10) < n, n-=b(s, q, 10); s--, r+=(os-s)*10^(q); os = s; q--)); r+= s; r}
\\ David A. Corneth, Jun 05 2016

from sympy.utilities.iterables import multiset_permutations
def auptodigs(maxdigits, b=10, sod=10): # works for any base, sum-of-digits
    alst = [sod] if 0 <= sod < b else []
    nzdigs = [i for i in range(1, b) if i <= sod]
    nzmultiset = []
    for d in range(1, b):
        nzmultiset += [d]*(sod//d)
    for d in range(2, maxdigits + 1):
        fullmultiset = [0]*(d-1-(sod-1)//(b-1)) + nzmultiset
        for firstdig in nzdigs:
            target_sum, restmultiset = sod - int(firstdig), fullmultiset[:]
            restmultiset.remove(firstdig)
            for p in multiset_permutations(restmultiset, d-1):
              if sum(p) == target_sum:
                  alst.append(int("".join(map(str, [firstdig]+p)), b))
                  if p[0] == target_sum:
                      break
    return alst
print(auptodigs(4)) # Michael S. Branicky, Sep 14 2021

def A052224(N = 19):
    """Return a generator of the sequence of all integers >= N with the same
    digit sum as N."""
    while True:
        yield N
        N = A228915(N) # skip to next larger integer with the same digit sum
a = A052224(); [next(a) for  in range(50)] # _M. F. Hasler, Mar 16 2022

a(n+1) = A228915(a(n)) for any n > 0. - Rémy Sigrist, Jul 10 2018

Incorrect formula deleted by N. J. A. Sloane, Jan 15 2009
Extended by Emeric Deutsch, Jan 16 2009
Offset changed by Bruno Berselli, Mar 07 2013

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

A052218 Numbers whose sum of digits is 4.

Original entry on oeis.org

4, 13, 22, 31, 40, 103, 112, 121, 130, 202, 211, 220, 301, 310, 400, 1003, 1012, 1021, 1030, 1102, 1111, 1120, 1201, 1210, 1300, 2002, 2011, 2020, 2101, 2110, 2200, 3001, 3010, 3100, 4000, 10003, 10012, 10021, 10030, 10102, 10111, 10120, 10201, 10210, 10300

Offset: 1

Henry Bottomley, Feb 01 2000

A007953(a(n)) = 4; number of repdigits = #{4,22,1111} = A242627(4) = 3. - Reinhard Zumkeller, Jul 17 2014

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1001 from Vincenzo Librandi and T. D. Noe, terms 1..201 from Vincenzo Librandi)

Cf. A007953.
Cf. A011557 (1), A052216 (2), A052217 (3), 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. A242614, A242627.

a052218 n = a052218_list !! (n-1)
a052218_list = filter ((== 4) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..10300] | &+Intseq(n) eq 4 ]; // Vincenzo Librandi, Mar 07 2013

Select[Range[10^5], Total[IntegerDigits[#]] == 4 &] (* Vincenzo Librandi, Mar 07 2013 *)
Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 11]], {s, IntegerPartitions[4]}]]] (* T. D. Noe, Mar 08 2013 *)

isok(n) = sumdigits(n) == 4; \\ Michel Marcus, Dec 28 2015

from itertools import count, islice
def agen(): yield from (10**i + 10**j + 10**k + 10**m for i in count(0) for j in range(i+1) for k in range(j+1) for m in range(k+1))
print(list(islice(agen(), 45))) # Michael S. Branicky, May 15 2022

Offset changed from Bruno Berselli, Mar 07 2013

A052221 Numbers whose sum of digits is 7.

Original entry on oeis.org

7, 16, 25, 34, 43, 52, 61, 70, 106, 115, 124, 133, 142, 151, 160, 205, 214, 223, 232, 241, 250, 304, 313, 322, 331, 340, 403, 412, 421, 430, 502, 511, 520, 601, 610, 700, 1006, 1015, 1024, 1033, 1042, 1051, 1060, 1105, 1114, 1123, 1132, 1141, 1150, 1204

Offset: 1

Henry Bottomley, Feb 01 2000

A007953(a(n)) = 7; number of repdigits = #{7,1111111} = A242627(7) = 2. - Reinhard Zumkeller, Jul 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Supersequence of A119461.
Cf. A007953, A242614, A242627.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), 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).

a052221 n = a052221_list !! (n-1)
a052221_list = filter ((== 7) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..1500] | &+Intseq(n) eq 7 ]; // Vincenzo Librandi, Mar 08 2013

Select[Range[1500],Total[IntegerDigits[#]]==7&] (* Harvey P. Dale, Apr 11 2012 *)

def ok(n): return sum(map(int, str(n))) == 7
print(list(filter(ok, range(1205)))) # Michael S. Branicky, Jul 16 2021

# faster version generating initial segment
from sympy.utilities.iterables import multiset_permutations
def auptodigs(maxdigits):
    alst = []
    for d in range(1, maxdigits+1):
        digset = "0"*(d-1) + "1111111222334567"
        for p in multiset_permutations(digset, d):
            if p[0] != '0' and sum(map(int, p)) == 7:
                alst.append(int("".join(p)))
    return alst
print(auptodigs(4)) # Michael S. Branicky, Jul 16 2021

Offset changed from Bruno Berselli, Mar 07 2013

A052223 Numbers whose sum of digits is 9.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 108, 117, 126, 135, 144, 153, 162, 171, 180, 207, 216, 225, 234, 243, 252, 261, 270, 306, 315, 324, 333, 342, 351, 360, 405, 414, 423, 432, 441, 450, 504, 513, 522, 531, 540, 603, 612, 621, 630, 702, 711, 720, 801, 810

Offset: 1

Henry Bottomley, Feb 01 2000

Any term of this sequence with an 11 appended cannot have 11 as prime factor. See A075154. [Lekraj Beedassy, Sep 27 2009]
A007953(a(n)) = 9; number of repdigits = #{9,333,1^9} = A242627(9) = 3. - Reinhard Zumkeller, Jul 17 2014
A010872(a(n)) = A010878(a(n)) = 0. - Ilya Gutkovskiy, Jun 04 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A007953.
Row n=9 of A245062.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A242614, A242627.

a052223 n = a052223_list !! (n-1)
a052223_list = filter ((== 9) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..1500] | &+Intseq(n) eq 9 ]; // Vincenzo Librandi, Mar 08 2013

Select[Range[1500], Total[IntegerDigits[#]] == 9 &] (* Vincenzo Librandi, Mar 08 2013 *)

More terms from Larry Reeves (Larryr(AT)acm.org), Sep 05 2000

Offset changed by Bruno Berselli, Mar 07 2013

A166311 Numbers whose sum of digits is 11.

Original entry on oeis.org

29, 38, 47, 56, 65, 74, 83, 92, 119, 128, 137, 146, 155, 164, 173, 182, 191, 209, 218, 227, 236, 245, 254, 263, 272, 281, 290, 308, 317, 326, 335, 344, 353, 362, 371, 380, 407, 416, 425, 434, 443, 452, 461, 470, 506, 515, 524, 533, 542, 551, 560, 605, 614

Offset: 1

Vincenzo Librandi, Oct 11 2009

A007953(a(n)) = 11; number of repdigits = A242627(11) = 1. - Reinhard Zumkeller, Jul 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A011557, A052216 - A052224, A143164, A166370, A166459.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A242614, A242627.

[n: n in [1..620] | &+Intseq(n) eq 11]; // Vincenzo Librandi, Mar 07 2013

Select[Range[620], Total[IntegerDigits[#]] == 11&] (* Vincenzo Librandi, Mar 07 2013 *)

Edited by N. J. A. Sloane, Oct 12 2009

A052219 Numbers whose sum of digits is 5.

Original entry on oeis.org

5, 14, 23, 32, 41, 50, 104, 113, 122, 131, 140, 203, 212, 221, 230, 302, 311, 320, 401, 410, 500, 1004, 1013, 1022, 1031, 1040, 1103, 1112, 1121, 1130, 1202, 1211, 1220, 1301, 1310, 1400, 2003, 2012, 2021, 2030, 2102, 2111, 2120, 2201, 2210, 2300, 3002

Offset: 1

Henry Bottomley, Feb 01 2000

A007953(a(n)) = 5; number of repdigits = #{5,11111} = A242627(5) = 2. - Reinhard Zumkeller, Jul 17 2014

Michael S. Branicky, Table of n, a(n) for n = 1..11628 (all terms through 15 digits; terms 1..1287 from Vincenzo Librandi and T. D. Noe, terms 1..462 from Vincenzo Librandi)

Cf. A007953.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), 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. A242614, A242627.

a052219 n = a052219_list !! (n-1)
a052219_list = filter ((== 5) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..3010] | &+Intseq(n) eq 5 ]; // Vincenzo Librandi, Mar 07 2013

Select[Range[10^4], Total[IntegerDigits[#]] == 5 &] (* Vincenzo Librandi, Mar 07 2013 *)
Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 9]], {s, IntegerPartitions[5]}]]] (* T. D. Noe, Mar 08 2013 *)

isok(n) = sumdigits(n) == 5; \\ Michel Marcus, Dec 28 2015

from sympy.utilities.iterables import multiset_permutations
def auptodigs(maxdigits):
  alst = [5]
  for d in range(2, maxdigits+1):
    fulldigset = list("0"*(d-1) + "1111122345")
    for firstdig in "12345":
      target_sum, restdigset = 5-int(firstdig), fulldigset[:]
      restdigset.remove(firstdig)
      for p in multiset_permutations(restdigset, d-1):
        if sum(map(int, p)) == target_sum:
          alst.append(int(firstdig+"".join(p)))
          if int(p[0]) == target_sum: break
  return alst
print(auptodigs(4)) # Michael S. Branicky, May 14 2021

Offset changed from Bruno Berselli, Mar 07 2013

A052222 Numbers whose sum of digits is 8.

Original entry on oeis.org

8, 17, 26, 35, 44, 53, 62, 71, 80, 107, 116, 125, 134, 143, 152, 161, 170, 206, 215, 224, 233, 242, 251, 260, 305, 314, 323, 332, 341, 350, 404, 413, 422, 431, 440, 503, 512, 521, 530, 602, 611, 620, 701, 710, 800, 1007, 1016, 1025, 1034, 1043, 1052, 1061

Offset: 1

Henry Bottomley, Feb 01 2000

A007953(a(n)) = 8; number of repdigits = #{8,44,2222,1^8} = A242627(8) = 4. - Reinhard Zumkeller, Jul 17 2014

Michael S. Branicky, Table of n, a(n) for n = 1..12870 (all terms <= 9 digits; terms 1..1000 from Vincenzo Librandi)

Cf. A007953.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), 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. A242614, A242627.

a052222 n = a052222_list !! (n-1)
a052222_list = filter ((== 8) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..1500] | &+Intseq(n) eq 8 ]; // Vincenzo Librandi, Mar 08 2013

Select[Range[1500], Total[IntegerDigits[#]] == 8 &] (* Vincenzo Librandi, Mar 08 2013 *)

from sympy.utilities.iterables import multiset_permutations
def auptodigs(maxdigits):
    alst = []
    for d in range(1, maxdigits+1):
        digset = "0"*(d-1) + "11111111222233445678"
        for p in multiset_permutations(digset, d):
            if p[0] != '0' and sum(map(int, p)) == 8:
                alst.append(int("".join(p)))
    return alst
print(auptodigs(4)) # Michael S. Branicky, Aug 17 2021

Offset changed from Bruno Berselli, Mar 07 2013

A052220 Numbers whose sum of digits is 6.

Original entry on oeis.org

6, 15, 24, 33, 42, 51, 60, 105, 114, 123, 132, 141, 150, 204, 213, 222, 231, 240, 303, 312, 321, 330, 402, 411, 420, 501, 510, 600, 1005, 1014, 1023, 1032, 1041, 1050, 1104, 1113, 1122, 1131, 1140, 1203, 1212, 1221, 1230, 1302, 1311, 1320, 1401, 1410

Offset: 1

Henry Bottomley, Feb 01 2000

A007953(a(n)) = 6; number of repdigits = #{6,33,222,111111} = A242627(6) = 4. - Reinhard Zumkeller, Jul 17 2014

There are binomial(t + 4, 5) terms having exactly t digits. Therefore binomial(t + 5, 6) have at most t digits. - David A. Corneth, Jun 29 2025

			1023 is in the sequence as it has digital sum 1 + 0 + 2 + 3 = 6. - _David A. Corneth_, Jun 29 2025

Michael S. Branicky, Table of n, a(n) for n = 1..12376 (terms 1..924 from Vincenzo Librandi; all terms with <= 12 digits)

Cf. A007953.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), 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. A242614, A242627.

a052220 n = a052220_list !! (n-1)
a052220_list = filter ((== 6) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..1500] | &+Intseq(n) eq 6 ]; // Vincenzo Librandi, Mar 07 2013

Select[Range[10^4], Total[IntegerDigits[#]] == 6 &] (* Vincenzo Librandi, Mar 07 2013 *)

nxt(n) = {my(v, c, toadd); v = valuation(n, 10); c = (n / 10^v)%10; toadd = 10^(v)*(10 - c) + c - 1; return(n + toadd)} \\ David A. Corneth, Jun 29 2025

from sympy.utilities.iterables import multiset_permutations
def auptodigs(maxdigits):
    alst = []
    for d in range(1, maxdigits+1):
        digset = "0"*(d-1) + "11111122233456"
        for p in multiset_permutations(digset, d):
            if p[0] != '0' and sum(map(int, p)) == 6:
                alst.append(int("".join(p)))
    return alst
print(auptodigs(4)) # Michael S. Branicky, Jun 15 2021

a(n) = a(n-1) + 10^v * (10 - c) + c-1 where c is the last nonzero digit of a(n) and v is the 10-adic valuation of a(n-1) and n > 1. - David A. Corneth, Jun 29 2025

Offset changed by Bruno Berselli, Mar 07 2013

cp's OEIS Frontend

A052216 Sums of two powers of 10.

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A052224 Numbers whose sum of digits is 10.

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A052217 Numbers whose sum of digits is 3.

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A052218 Numbers whose sum of digits is 4.

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A052221 Numbers whose sum of digits is 7.

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