A138530 -id:A138530 - cp's OEIS frontend

A053832 Sum of digits of n written in base 12.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 7, 8

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8,9,10,11}, 1->{1,2,3,4,5,6,7,8,9,10,11,12}, 2->{2,3,4,5,6,7,8,9,10,11,12,13}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 1 + 8 = 9 because 20 is written as 18 base 12.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Duodecimal.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A000120, A007953, A064459, A138530.

a053832 n = q 0 $ divMod n 12 where
   q r (0, d) = r + d
   q r (m, d) = q (r + d) $ divMod m 12
-- Reinhard Zumkeller, May 15 2011

Table[Plus @@ IntegerDigits[n, 12], {n, 0, 85}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 11}]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)

a(n)=if(n<1,0,if(n%12,a(n-1)+1,a(n/12)))

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(12n+i) = a(n)+i for 0 <= i <= 11.
a(n) = n-11*(Sum_{k>0} floor(n/12^k)) = n-11*A064459(n). (End)
a(n) = A138530(n,12) for n > 11. - Reinhard Zumkeller, Mar 26 2008
Sum_{n>=1} a(n)/(n*(n+1)) = 12*log(12)/11 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A053831 Sum of digits of n written in base 11.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 7, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8,9,10}, 1->{1,2,3,4,5,6,7,8,9,10,11}, 2->{2,3,4,5,6,7,8,9,10,11,12}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 1 + 9 = 10 because 20 is written as 19 base 11.

Tanar Ulric, Table of n, a(n) for n = 0..10000
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A007953, A064458, A138530.

Sum of digits of n written in bases 2-16: A000120, A053735, A053737, A053824, A053827, A053828, A053829, A053830, A007953, this sequence, A053832, A053833, A053834, A053835, A053836.

int Base11DigitSum(int n) {
   int count = 0;
   while (n != 0) { count += n % 11; n = n / 11; }
   return count;
} // Tanar Ulric, Oct 20 2021

Table[Plus @@ IntegerDigits[n, 11], {n, 0, 86}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 10}]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)

a(n)=if(n<1,0,if(n%11,a(n-1)+1,a(n/11)))

a(n)=sumdigits(n,11) \\ Charles R Greathouse IV, Oct 20 2021

From Benoit Cloitre, Dec 19 2002: (Start)
a(0)=0, a(11n+i) = a(n)+i for 0 <= i <= 10.
a(n) = n-(m-1)*(Sum_{k>0} floor(n/m^k)) = n-(m-1)*A064458(n). (End)
a(n) = A138530(n,11) for n > 10. - Reinhard Zumkeller, Mar 26 2008
Sum_{n>=1} a(n)/(n*(n+1)) = 11*log(11)/10 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A053833 Sum of digits of n written in base 13.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 6, 7, 8, 9, 10, 11, 12

Offset: 0

Henry Bottomley, Mar 28 2000

			a(20) = 1 + 7 = 8 because 20 is written as "17" in base 13.

Tanar Ulric, Table of n, a(n) for n = 0..10000
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A000120, A007953, A138530.

Total[IntegerDigits[#,13]]&/@Range[0,90]  (* Harvey P. Dale, Jul 17 2012 *)

a(n)=if(n<1,0,if(n%13,a(n-1)+1,a(n/13)))

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(13n+i) = a(n)+i for 0 <= i <= 12.
a(n) = n-12*(Sum_{k>0} floor(n/13^k)). (End)
a(n) = A138530(n,13) for n > 12. - Reinhard Zumkeller, Mar 26 2008
Sum_{n>=1} a(n)/(n*(n+1)) = 13*log(13)/12 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A053834 Sum of digits of n written in base 14.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

Offset: 0

Henry Bottomley, Mar 28 2000

			a(20) = 1 + 6 = 7 because 20 is written as "16" in base 14.

Tanar Ulric, Table of n, a(n) for n = 0..10000
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A000120, A007953, A138530.

Array[Total[IntegerDigits[#,14]]&,90,0] (* Harvey P. Dale, Jul 16 2011 *)

a(n)=if(n<1,0,if(n%14,a(n-1)+1,a(n/14)))

a(n) = sumdigits(n, 14); \\ Michel Marcus, Jun 03 2021

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(14n+i) = a(n)+i for 0 <= i <= 13.
a(n) = n-13*(Sum_{k>0} floor(n/14^k)). (End)
a(n) = A138530(n,14) for n > 13. - Reinhard Zumkeller, Mar 26 2008
Sum_{n>=1} a(n)/(n*(n+1)) = 14*log(14)/13 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A053835 Sum of digits of n written in base 15.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 5, 6, 7, 8, 9, 10, 11, 12, 13

Offset: 0

Henry Bottomley, Mar 28 2000

			a(20) = 1 + 5 = 6 because 20 is written as "15" in base 15.

Tanar Ulric, Table of n, a(n) for n = 0..10000
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A000120, A007953, A138530.

a[n_] := Total[IntegerDigits[n, 15]]; Array[a, 100, 0] (* Amiram Eldar, Aug 03 2023 *)

a(n)=if(n<1,0,if(n%15,a(n-1)+1,a(n/15)))

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(15n+i) = a(n)+i, 0<=i<=14.
a(n) = n - 14*(Sum_{k>0} floor(n/15^k)). (End)
a(n) = A138530(n,15) for n > 14. - Reinhard Zumkeller, Mar 26 2008
Sum_{n>=1} a(n)/(n*(n+1)) = 15*log(15)/14 (Shallit, 1984). - Amiram Eldar, Aug 03 2023

A240236 Triangle read by rows: sum of digits of n in base k, for 2<=k<=n.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 3, 2, 1, 3, 3, 4, 3, 2, 1, 1, 4, 2, 4, 3, 2, 1, 2, 1, 3, 5, 4, 3, 2, 1, 2, 2, 4, 2, 5, 4, 3, 2, 1, 3, 3, 5, 3, 6, 5, 4, 3, 2, 1, 2, 2, 3, 4, 2, 6, 5, 4, 3, 2, 1, 3, 3, 4, 5, 3, 7, 6, 5, 4, 3, 2, 1, 3, 4, 5, 6, 4, 2, 7, 6, 5, 4, 3, 2, 1

Offset: 2

Franklin T. Adams-Watters, Apr 02 2014

			Triangle starts:
  1
  2 1
  1 2 1
  2 3 2 1
  2 2 3 2 1
  3 3 4 3 2 1

Reinhard Zumkeller, Rows n = 2..100 of triangle, flattened

Row sums give A043306.
Cf. A000120, A053735, A053737, A007953.
See A138530 for another version.
Cf. A002260, A090623.

a240236 n k = a240236_tabl !! (n-1) !! (k-1)
a240236_row n = a240236_tabl !! (n-1)
a240236_tabl = zipWith (map . flip q)
                       [2..] (map tail $ tail a002260_tabl) where
   q b n = if n < b then n else q b n' + d where (n', d) = divMod n b
-- Reinhard Zumkeller, Apr 29 2015

Table[Total[Flatten[IntegerDigits[n,k]]],{n,20},{k,2,n}]//Flatten (* Harvey P. Dale, Jan 13 2025 *)

T(n,k) = local(r=0);if(k<2,-1,while(n>0,r+=n%k;n\=k);r)

T(n, k) = sumdigits(n, k) \\ Zhuorui He, Aug 25 2025

T(n,k) = n - (k - 1) * Sum_{i=1..floor(log_k(n))} floor(n/k^i). - Ridouane Oudra, Sep 27 2024

T(n,k) = n - (k - 1) * A090623(n,k). - Zhuorui He, Aug 25 2025

A131383 Total digital sum of n: sum of the digital sums of n for all the bases 1 to n (a 'digital sumorial').

Original entry on oeis.org

1, 3, 6, 8, 13, 16, 23, 25, 30, 35, 46, 46, 59, 66, 75, 74, 91, 91, 110, 112, 125, 136, 159, 152, 169, 182, 195, 199, 228, 223, 254, 253, 274, 291, 316, 297, 334, 353, 378, 373, 414, 409, 452, 460, 475, 498, 545, 520, 557, 565, 598, 608, 661, 652, 693, 690

Offset: 1

Hieronymus Fischer, Jul 05 2007, Jul 15 2007, Jan 07 2009

Sums of rows of the triangle in A138530. - Reinhard Zumkeller, Mar 26 2008

			5 = 11111(base 1) = 101(base 2) = 12(base 3) = 11(base 4) = 10(base 5). Thus a(5) = ds_1(5)+ds_2(5)+ds_3(5)+ds_4(5)+ds_5(5) = 5+2+3+2+1 = 13.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Sum

Cf. A131384, A007953.

Table[n + Total@ Map[Total@ IntegerDigits[n, #] &, Range[2, n]], {n, 56}] (* Michael De Vlieger, Jan 03 2017 *)

a(n)=sum(i=2,n+1,vecsum(digits(n,i))); \\ R. J. Cano, Jan 03 2017

a(n) = n^2-sum{k>0, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

a(n) = n^2-sum{2<=p<=n, (p-1)*sum{0

a(n) = n^2-A024916(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

a(n) = A004125(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

Asymptotic behavior: a(n) = (1-Pi^2/12)*n^2 + O(n*log(n)) = A004125(n) + A006218(n) + O(n*log(n)).

Lim a(n)/n^2 = 1 - Pi^2/12 for n-->oo.

G.f.: (1/(1-x))*(x(1+x)/(1-x)^2-sum{k>0,sum{j>1,(j-1)*x^(j^k)/(1-x^(j^k))}= }).

Also: (1/(1-x))*(x(1+x)/(1-x)^2-sum{m>1, sum{10,j^(1/k) is an integer, j^(1/k)-1}}*x^m}).

a(n) = n^2-sum{10,sum{1

Recurrence: a(n)=a(n-1)-b(n)+2n-1, where b(n)=sum{1

a(n) = sum{1<=p<=n, ds_p(n)} where ds_p = digital sum base p.

a(n) = A043306(n) + n (that sequence ignores unary) = A014837(n) + n + 1 (that sequence ignores unary and base n in which n is "10"). - Alonso del Arte, Mar 26 2009

A339541 a(n) = n + sod(n, sod(n, 10)), where sod(n,b) is the sum of base-b digits of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 14, 14, 17, 20, 20, 20, 20, 20, 29, 22, 24, 26, 30, 28, 32, 31, 30, 38, 38, 32, 38, 36, 41, 44, 42, 40, 47, 46, 45, 44, 46, 44, 50, 53, 50, 56, 54, 52, 62, 52, 57, 56, 64, 60, 65, 62, 70, 68, 66, 65, 68, 75, 70, 74, 80, 77, 74, 84, 82, 74, 79, 80, 83, 88, 84, 92

Offset: 1

Robert Israel, Dec 08 2020

			a(15) = 15 + 5 = 20 because sod(15,10) = 6 and sod(15,6) = 5 (because 15 = 23_6).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007953, A138530

sod:= proc(x,b) if b=1 then x else convert(convert(x,base,b),`+`) fi end proc:
f:= x -> x + sod(x,sod(x,10)):
map(f, [$1..100]);

a(n) = n + A138530(n, A007953(n)).

A339542 Primes p such that A339541(p) is prime.

Original entry on oeis.org

2, 13, 19, 37, 71, 73, 127, 163, 167, 181, 271, 293, 307, 367, 431, 433, 457, 503, 569, 631, 659, 811, 907, 983, 1009, 1087, 1153, 1171, 1229, 1373, 1399, 1409, 1423, 1483, 1487, 1511, 1597, 1777, 1801, 1861, 1867, 1999, 2017, 2039, 2053, 2143, 2239, 2273, 2297, 2341, 2383, 2437, 2477, 2521, 2659

Offset: 1

J. M. Bergot and Robert Israel, Dec 08 2020

Primes p such that p + A138530(p, A007953(p)) is prime.

			a(5) = 71 is in the sequence because sod(71,10) = 8, sod(71,8) = 8 (since 71 = 107_8), and 71+8=79 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007953, A138530, A339541.

sod:= proc(x,b) if b=1 then x else convert(convert(x,base,b),`+`) fi end proc:
select(p -> isprime(p+sod(p,sod(p,10))), [seq(ithprime(i),i=1..1000)]);

A356517 Square array A(n, k), n >= 2, k >= 0, read by antidiagonals upwards; A(n, k) is the least integer with sum of digits k in base n.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 2, 7, 0, 1, 2, 5, 15, 0, 1, 2, 3, 8, 31, 0, 1, 2, 3, 7, 17, 63, 0, 1, 2, 3, 4, 11, 26, 127, 0, 1, 2, 3, 4, 9, 15, 53, 255, 0, 1, 2, 3, 4, 5, 14, 31, 80, 511, 0, 1, 2, 3, 4, 5, 11, 19, 47, 161, 1023, 0, 1, 2, 3, 4, 5, 6, 17, 24, 63, 242, 2047

Offset: 2

Rémy Sigrist, Aug 10 2022

The expansion of A(n, k) in base n is:
       q  n-1  ...  n-1
          <- p times ->
where q = k mod (n-1) and p = floor(k / (n-1)).

			Array A(n, k) begins:
  n\k|  0  1  2  3   4   5   6    7    8    9    10    11    12
  ---+---------------------------------------------------------
    2|  0  1  3  7  15  31  63  127  255  511  1023  2047  4095
    3|  0  1  2  5   8  17  26   53   80  161   242   485   728
    4|  0  1  2  3   7  11  15   31   47   63   127   191   255
    5|  0  1  2  3   4   9  14   19   24   49    74    99   124
    6|  0  1  2  3   4   5  11   17   23   29    35    71   107
    7|  0  1  2  3   4   5   6   13   20   27    34    41    48
    8|  0  1  2  3   4   5   6    7   15   23    31    39    47
    9|  0  1  2  3   4   5   6    7    8   17    26    35    44
   10|  0  1  2  3   4   5   6    7    8    9    19    29    39
Array A(n, k) begins (with values given in base n):
  n\k|  0  1   2    3     4      5       6        7         8          9
  ---+------------------------------------------------------------------
    2|  0  1  11  111  1111  11111  111111  1111111  11111111  111111111
    3|  0  1   2   12    22    122     222     1222      2222      12222
    4|  0  1   2    3    13     23      33      133       233        333
    5|  0  1   2    3     4     14      24       34        44        144
    6|  0  1   2    3     4      5      15       25        35         45
    7|  0  1   2    3     4      5       6       16        26         36
    8|  0  1   2    3     4      5       6        7        17         27
    9|  0  1   2    3     4      5       6        7         8         18
   10|  0  1   2    3     4      5       6        7         8          9

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 antidiagonals)

Cf. A000225, A051885, A062318, A140576, A165804, A180516, A181287, A181288, A181303, A138530, A240236.

PARI
```
A(n,k) = { (1+k%(n-1))*n^(k\(n-1))-1 }
```

def A(n,k): return (1+(k % (n-1)))*n**(k//(n-1))-1

A(2, k) = 2^k - 1.
A(3, k) = A062318(k+1).
A(4, k) = A180516(k+1).
A(5, k) = A181287(k+1).
A(6, k) = A181288(k+1).
A(7, k) = A181303(k+1).
A(8, k) = A165804(k+1).
A(9, k) = A140576(k+1).
A(10, k) = A051885(k).
A(n, 0) = 0.
A(n, 1) = 1.
A(n, k) = k iff k < n.
A(n, n) = 2*n - 1.
A(n, n+1) = 3*n - 1 for any n > 2.

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cp's OEIS Frontend

A053832 Sum of digits of n written in base 12.

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A053831 Sum of digits of n written in base 11.

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A053833 Sum of digits of n written in base 13.

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A053834 Sum of digits of n written in base 14.

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A053835 Sum of digits of n written in base 15.

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A240236 Triangle read by rows: sum of digits of n in base k, for 2<=k<=n.

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A131383 Total digital sum of n: sum of the digital sums of n for all the bases 1 to n (a 'digital sumorial').

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