A043306 -id:A043306 - cp's OEIS frontend

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

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

A043000 Number of digits in all base-b representations of n, for 2 <= b <= n.

Original entry on oeis.org

2, 4, 7, 9, 11, 13, 16, 19, 21, 23, 25, 27, 29, 31, 35, 37, 39, 41, 43, 45, 47, 49, 51, 54, 56, 59, 61, 63, 65, 67, 70, 72, 74, 76, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126

Offset: 2

Clark Kimberling

From A.H.M. Smeets, Dec 14 2019: (Start)
a(n)-a(n-1) >= 2 due to the fact that n = 10_n, so there is an increment of at least 2. If n can be written as a perfect power m^s, an additional +1 comes to it for the representation of n in each base m.
For instance, for n = 729 we have 729 = 3^6 = 9^3 = 27^2, so there is an additional increment of 3. For n = 1296 we have 1296 = 6^4 = 36^2, so there is an additional increment of 2. For n = 4096 we have 4096 = 2^12 = 4^6 = 8^4 = 16^3= 64^2, so there is an additional increment of 5. (End)

			5 = 101_2 = 12_3 = 11_4 = 10_5. Thus a(5) = 3+2+2+2 = 9.

A.H.M. Smeets, Table of n, a(n) for n = 2..20000
Vaclav Kotesovec, Plot of a(n)/(2*n) for n = 2..1000000

Cf. A001597, A043306, A068953, A191322, A253642, A255165.

[&+[Floor(Log(i,i*n)):k in [2..n]]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019

A043000 := proc(n) add( nops(convert(n,base,b)),b=2..n) ; end proc: # R. J. Mathar, Jun 04 2011

Table[Total[IntegerLength[n,Range[2,n]]],{n,2,60}] (* Harvey P. Dale, Apr 23 2019 *)

a(n)=sum(b=2,n,#digits(n,b)) \\ Jeppe Stig Nielsen, Dec 14 2019

a(n)= n-1 +sum(b=2,n,logint(n,b)) \\ Jeppe Stig Nielsen, Dec 14 2019

a(n) = {2*n-2+sum(i=2, logint(n, 2), sqrtnint(n, i)-1)} \\ David A. Corneth, Dec 31 2019

first(n) = my(res = vector(n)); res[1] = 2; for(i = 2, n, inc = numdiv(gcd(factor(i+1)[,2]))+1; res[i] = res[i-1]+inc); res \\ David A. Corneth, Dec 31 2019

def count(n,b):
    c = 0
    while n > 0:
        n, c = n//b, c+1
    return c
n = 0
while n < 50:
    n = n+1
    a, b = 0, 1
    while b < n:
        b = b+1
        a = a + count(n,b)
    print(n,a) # A.H.M. Smeets, Dec 14 2019

a(n) = Sum_{i=2..n} floor(log_i(i*n)); a(n) ~ 2*n. - Vladimir Shevelev, Jun 03 2011 [corrected by Vaclav Kotesovec, Apr 05 2021]
a(n) = A070939(n) + A081604(n) + A110591(n) + ... + 1. - R. J. Mathar, Jun 04 2011
From Ridouane Oudra, Nov 13 2019: (Start)
a(n) = Sum_{i=1..n-1} floor(n^(1/i));
a(n) = n - 1 + Sum_{i=1..floor(log_2(n))} floor(n^(1/i) - 1);
a(n) = n - 1 + A255165(n). (End)
If n is in A001597 then a(A001597(m)) - a(A001597(m)-1) = 2 + A253642(m), otherwise a(n) - a(n-1) = 2. - A.H.M. Smeets, Dec 14 2019

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

A309531 a(n) = Sum_{k=2..n} (-1)^k * A240236(n, k).

Original entry on oeis.org

1, 1, 0, 0, 2, 2, -3, 1, 5, 5, -2, -2, 4, 10, -6, -6, 1, 1, -8, 0, 10, 10, -16, -8, 4, 18, 7, 7, 13, 13, -19, -7, 9, 19, -11, -11, 7, 21, -15, -15, -5, -5, -20, 10, 32, 32, -36, -24, -1, 17, 0, 0, 18, 32, -14, 6, 34, 34, -23, -23, 7, 45, -29, -13, 5, 5, -16, 8, 30, 30, -66

Offset: 2

Seiichi Manyama, Aug 06 2019

			a(2) = 1.
a(3) = 2 - 1 = 1.
a(4) = 1 - 2 + 1 = 0.
a(5) = 2 - 3 + 2 - 1 = 0.
a(6) = 2 - 2 + 3 - 2 + 1 = 2.

Seiichi Manyama, Table of n, a(n) for n = 2..1000

Cf. A000040, A043306, A065091, A240236.

{a(n) = sum(k=2, n, (-1)^k*sumdigits(n, k))}

a(n) = a(n-1) if and only if n is an odd prime.

A043544 Numbers k such that s(k)

Original entry on oeis.org

12, 16, 18, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 90, 96, 100, 102, 108, 112, 114, 120, 126, 128, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220

Offset: 1

Clark Kimberling

Cf. A043306.

A = {k: A043306(k) < A043306(k - 1)}. - Sean A. Irvine, Mar 09 2021

A244912 Sum of leading digits in representations of n in bases 2,3,...,n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 9, 11, 12, 15, 16, 18, 20, 20, 21, 25, 26, 29, 31, 33, 34, 38, 36, 38, 39, 42, 43, 47, 48, 52, 54, 56, 58, 58, 59, 61, 63, 67, 68, 72, 73, 76, 79, 81, 82, 88, 84, 88, 90, 93, 94, 99, 101, 105, 107, 109, 110, 116, 117, 119, 122, 117, 119, 123

Offset: 2

Alex Ratushnyak, Jul 08 2014

			8 in bases 2...8 is:
  1000 (base 2)
  22 (base 3)
  20 (base 4)
  13 (base 5)
  12 (base 6)
  11 (base 7)
  10 (base 8)
The sum of first digits is 1+2+2+1+1+1+1 = 9, so a(8)=9.

Jens Kruse Andersen, Table of n, a(n) for n = 2..1000

Cf. A004125 (sum of last digits), A043306 (sum of all digits).

f[n_] := Sum[ IntegerDigits[n, k][[1]], {k, 2, n}]; Array[f, 70, 2] (* Robert G. Wilson v, Aug 02 2014 *)

a(n) = sum(i=2, n, digits(n, i)[1]); \\ Michel Marcus, Jul 17 2014

import math
def modlg(a, b):
    return a // b**int(math.log(a, b))
for n in range(2,77):
    s=0
    for k in range(2,n+1):
        s += modlg(n,k)
    print(s, end=', ')

a(n) = Sum_{k=2..n} floor(n/f(n,k)), with f(n,k) = k^floor(log_k(n)). - Ridouane Oudra, Apr 26 2025

A343481 a(n) is the sum of all digits of n in every prime base 2 <= p <= n.

Original entry on oeis.org

1, 3, 3, 6, 6, 10, 11, 11, 10, 15, 16, 22, 21, 21, 23, 30, 32, 40, 42, 42, 39, 48, 52, 53, 49, 52, 53, 63, 66, 77, 83, 82, 76, 77, 82, 94, 87, 85, 90, 103, 107, 121, 123, 129, 120, 135, 144, 147, 153, 150, 151, 167, 176, 178, 185, 181, 168, 185, 194, 212, 199

Offset: 2

Amiram Eldar, Apr 16 2021

			a(5) = 6 since in the prime bases 2, 3 and 5 the representations of 5 are 101_2, 12_3 and 10_5, respectively, and (1 + 0 + 1) + (1 + 2) + (1 + 0) = 6.

Amiram Eldar, Table of n, a(n) for n = 2..10001
Robin Fissum, Digit sums and the number of prime factors of the factorial n!=1.2...n, Bachelor's project in BMAT, Norwegian University of Science and Technology, 2020; ResearchGate link.

Cf. A014837, A043306, A072691, A073002, A222171.

s[n_, b_] := Plus @@ IntegerDigits[n, b]; ps[n_] := Select[Range[n], PrimeQ]; a[n_] := Sum[s[n, b], {b, ps[n]}]; Array[a, 100, 2]

a(n) = sum(b=2, n, if (isprime(b), sumdigits(n, b))); \\ Michel Marcus, Apr 17 2021

a(n) ~ (1-Pi^2/12)*n^2/log(n) + c*n^2/log(n)^2 + o(n^2/log(n)^2), where c = 1 - Pi^2/24 + zeta'(2)/2 = 1 - A222171 - (1/2)*A073002 = 0.1199923561... (Fissum, 2020).

Showing 1-8 of 8 results.

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A014837 Sum of all the digits of n in every base from 2 to n-1.

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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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A043000 Number of digits in all base-b representations of n, for 2 <= b <= 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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A309531 a(n) = Sum_{k=2..n} (-1)^k * A240236(n, k).

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A043544 Numbers k such that s(k)

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A244912 Sum of leading digits in representations of n in bases 2,3,...,n.

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