A240236 -id:A240236 - cp's OEIS frontend

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

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.

A309431 a(n) = Sum_{k=2..p} (-1)^k * A240236(p, k), where p is n-th prime.

Original entry on oeis.org

1, 1, 0, 2, 5, -2, -6, 1, 10, 7, 13, -11, -15, -5, 32, 0, 34, -23, 5, 30, -66, 33, 29, -18, -60, -17, 13, 69, -21, -65, 65, 17, 8, 50, -6, 24, -33, -23, 130, 13, 115, -93, 184, -132, 4, 48, -11, 31, 70, -99, -23, 192, -258, 154, -68, 155, -18, 16, -35, -101, -11

Offset: 1

Seiichi Manyama, Aug 06 2019

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000040, A240236, A309531.

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

a(n) = A309531(A000040(n)).

A138530 Triangle read by rows: T(n,k) = sum of digits of n in base k representation, 1<=k<=n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 26 2008

A131383(n) = sum of n-th row;
A000027(n) = T(n,1);
A000120(n) = T(n,2) for n>1;
A053735(n) = T(n,3) for n>2;
A053737(n) = T(n,4) for n>3;
A053824(n) = T(n,5) for n>4;
A053827(n) = T(n,6) for n>5;
A053828(n) = T(n,7) for n>6;
A053829(n) = T(n,8) for n>7;
A053830(n) = T(n,9) for n>8;
A007953(n) = T(n,10) for n>9;
A053831(n) = T(n,11) for n>10;
A053832(n) = T(n,12) for n>11;
A053833(n) = T(n,13) for n>12;
A053834(n) = T(n,14) for n>13;
A053835(n) = T(n,15) for n>14;
A053836(n) = T(n,16) for n>15;
A007395(n) = T(n,n-1) for n>1;
A000012(n) = T(n,n).

			Start of the triangle for n in base k representation:
......................1
....................11....10
......... ........111....11...10
................1111...100...11..10
..............11111...101...12..11..10
............111111...110...20..12..11..10
..........1111111...111...21..13..12..11..10
........11111111..1000...22..20..13..12..11..10
......111111111..1001..100..21..14..13..12..11..10
....1111111111..1010..101..22..20..14..13..12..11..10
..11111111111..1011..102..23..21..15..14..13..12..11..10
111111111111..1100..110..30..22..20..15..14..13..12..11..10,
and the triangle of sums of digits starts:
......................1
.....................2...1
......... ..........3...2...1
...................4...1...2...1
..................5...2...3...2...1
.................6...2...2...3...2...1
................7...3...3...4...3...2...1
...............8...1...4...2...4...3...2...1
..............9...2...1...3...5...4...3...2...1
............10...2...2...4...2...5...4...3...2...1
...........11...3...3...5...3...6...5...4...3...2...1
..........12...2...2...3...4...2...6...5...4...3...2...1.

Alois P. Heinz, Rows n = 1..141, flattened
Eric Weisstein's World of Mathematics, Digit Sum

Cf. A007953. See A240236 for another version.

Cf. A002260.

a138530 n k = a138530_tabl !! (n-1) !! (k-1)
a138530_row n = a138530_tabl !! (n-1)
a138530_tabl = zipWith (map . flip q) [1..] a002260_tabl where
   q 1 n = n
   q b n = if n < b then n else q b n' + d where (n', d) = divMod n b
-- Reinhard Zumkeller, Apr 29 2015

T[n_, k_] := If[k == 1, n, Total[IntegerDigits[n, k]]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 25 2021 *)

A144912 Unreduced numerators of digital mean, dm_num(b, n), with rows n in {2, 3, 4, ...} and columns b in {2, 3, 4, ..., n}.

Original entry on oeis.org

0, 2, -2, -1, 0, -4, 1, 2, -2, -6, 1, 0, 0, -4, -8, 3, 2, 2, -2, -6, -10, -2, 4, -2, 0, -4, -8, -12, 0, -4, 0, 2, -2, -6, -10, -14, 0, -2, 2, -4, 0, -4, -8, -12, -16, 2, 0, 4, -2, 2, -2, -6, -10, -14, -18, 0, -2, 0, 0, -6, 0, -4, -8, -12, -16, -20

Offset: 2

Reikku Kulon, Sep 25 2008, Oct 03 2008

The unreduced numerator of dm(b, n) is Sum_{i=1..d} (2*d_i - (b-1)), where d is the number of digits in the base b representation of n and d_i the individual digits. The corresponding denominator is 2 * d, giving a value in (-(b - 1) / 2, (b - 1) / 2] for n > 0.
dm_num(b, n) = d(b - 1) iff all the digits in n are b - 1.
dm_num(b, n) = -2(b - 2) for b = n, because n in base n is 10, giving dm_num(n, n) = 2 - n + 1 + 0 - n + 1 = 4 - 2 * n = -2(n - 2).
dm_num(b, n) = 0 for odd b and n having all digits equal to (b - 1) / 2, as well as for many other (b, n).
Defining m = ceiling((n + 1) / 2):
dm_num(b, n) = dm_num(b - 1, n) - 4 for b in [m + 1, n].
dm_num(m, n) = 0 for even n and 2 for odd n.
dm_num(m - 1, n) = 6 - n for even n > 4 and 9 - n for odd n > 5, producing a sequence of first differences {+2, -4, +2, -4, ...}.
Triangular patterns become clearly visible for large n, defined by additive periodicities along rational slopes. Zeros along the triangle borders correspond to ones in the Redheffer matrix until odd values become dominant. The line along m is the border between the two largest triangles. This pattern is masked by aliasing effects for small bases, notably including base 10, due to the thinness of the triangles which dominate at small b. Odd values may represent "artifacts" caused by "interference".

			Triangle begins:
   0;
   2, -2;
  -1,  0, -4;
   1,  2, -2, -6;
   1,  0,  0, -4, -8;
   3,  2,  2, -2, -6, -10;
   ...

Reikku Kulon, Rows of triangle for b in [2, 142]
Reikku Kulon, C99 source to produce the triangle
Reikku Kulon, Triangle as 2048x2048 PNG image (zero is white, primes are black and darker grays indicate fewer prime factors)
Reikku Kulon, Triangle as 2048x2048 PNG image, extended to b in [2, 2 * n + 1]
Reikku Kulon, Prime band as 16384x256 PNG image (note the curves coincident with new strips of primes, as well as the second band which appears at 4096 and corresponds to the 637/638 gap in A031443)
Reikku Kulon, Prime band as 16384x256 PNG image, starting from n = 57344
Eric Weisstein's World of Mathematics, Redheffer Matrix

Cf. A002321, A031443, A083058, A144777, A144798, A144799, A144800, A144801, A144812, A144923, A240236.

dmnum[b_,n_]:=2Total[IntegerDigits[n,b]]-(b-1)Floor[Log[b,n*b]]; (* after Jinyuan Wang *)
Table[dmnum[b,n],{n,2,10},{b,2,n}] (* Paolo Xausa, Sep 26 2023 *)

dm(b, n) = 2*sumdigits(n, b) - (b-1)*logint(n*b, b); \\ Jinyuan Wang, Jul 21 2020

A043306 Sum of all digits in all base-b representations for n, for 2 <= b <= n.

Original entry on oeis.org

1, 3, 4, 8, 10, 16, 17, 21, 25, 35, 34, 46, 52, 60, 58, 74, 73, 91, 92, 104, 114, 136, 128, 144, 156, 168, 171, 199, 193, 223, 221, 241, 257, 281, 261, 297, 315, 339, 333, 373, 367, 409, 416, 430, 452, 498, 472, 508, 515, 547, 556, 608, 598, 638, 634, 670, 698, 756, 717, 777

Offset: 2

Clark Kimberling

			5 = 101_2 = 12_3 = 11_4 = 10_5. Thus a(5) = 2 + 3 + 2 + 1 = 8.

Alois P. Heinz, Table of n, a(n) for n = 2..1000
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.
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme, Math. Semesterber., Vol. 49, No. 2 (2002), pp. 209-226; alternative link.
Vladimir Shevelev, Compact integers and factorials, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (see p. 205).

Cf. A014837, A043000, A068953, A191251, A191322, A191350, A309531.

Row sums of A240236.

Table[Sum[Total[First[RealDigits[n, i]]], {i, 2, n}], {n, 2, 80}] (* Carl Najafi, Aug 16 2011 *)

a(n) = sum(i=2, n, vecsum(digits(n, i))); \\ Michel Marcus, Jan 03 2017

a(n) = sum(b=2, n, sumdigits(n, b)); \\ Michel Marcus, Aug 18 2017

from sympy.ntheory.digits import digits
def a(n): return sum(sum(digits(n, b)[1:]) for b in range(2, n+1))
print([a(n) for n in range(2, 62)]) # Michael S. Branicky, Apr 04 2022

From Vladimir Shevelev, Jun 03 2011: (Start)
a(n) = (n-1)*n - Sum_{i=2..n} (i-1)*Sum_{r>=1} floor(n/i^r).
a(n) <= (n-1)^2*log(n+1)/log(n).
Problem: find a better upper estimate. (End)
From Amiram Eldar, Apr 16 2021: (Start)
a(n) = A014837(n) + 1.
a(n) ~ (1-Pi^2/12)*n^2 + O(n^(3/2)) (Fissum, 2020). (End)

A090623 Triangle of T(n,k) = [n/k] + [n/k^2] + [n/k^3] + [n/k^4] + ... for n, k > 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 4, 2, 1, 1, 1, 4, 2, 1, 1, 1, 1, 7, 2, 2, 1, 1, 1, 1, 7, 4, 2, 1, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 11, 5, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 11, 6, 3, 3, 2, 2, 1, 1, 1

Offset: 2

Henry Bottomley, Dec 06 2003

			Rows start:
  1;
  1,1;
  3,1,1;
  3,1,1,1;
  4,2,1,1,1;
  4,2,1,1,1,1;
  7,2,2,1,1,1,1;
  7,4,2,1,1,1,1,1;
  8,4,2,2,1,1,1,1,1;
  ...

Zhuorui He, Table of n, a(n) for n = 2..11326
Wenguang Zhai, On the prime power factorization of n!, Journal of Number Theory, Volume 129, Issue 8, August 2009, Pages 1820-1836.

Columns include A011371, A054861, A054893, A027868, A054895, A054896, A054897, A054898, A054899, A064458, A064459, A090620, A054900.
Row sums: A078632.
Cf. A240236.

A090623[n_, k_] := Quotient[n - DigitSum[n, k], k - 1];
Table[A090623[n, k], {n, 2, 15}, {k, 2, n}] (* Paolo Xausa, Sep 02 2025 *)

T(n,k) = {my(s = 0, j = 1); while(p=n\k^j, s += p; j++); s;} \\ Michel Marcus, Feb 02 2016

T(n,k) = (n - sumdigits(n,k))/(k-1) \\ Zhuorui He, Aug 25 2025

For p prime, T(n, p) = A090622(n, p) is the number of times that p is a factor of n!.

T(n,k) = (n - A240236(n, k))/(k - 1). - Zhuorui He, Aug 25 2025

a(41) onward corrected by Zhuorui He, Aug 25 2025

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.

cp's OEIS Frontend

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

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A309431 a(n) = Sum_{k=2..p} (-1)^k * A240236(p, k), where p is n-th prime.

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A138530 Triangle read by rows: T(n,k) = sum of digits of n in base k representation, 1<=k<=n.

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Haskell

Mathematica

A144912 Unreduced numerators of digital mean, dm_num(b, n), with rows n in {2, 3, 4, ...} and columns b in {2, 3, 4, ..., n}.

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A043306 Sum of all digits in all base-b representations for n, for 2 <= b <= n.

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A090623 Triangle of T(n,k) = [n/k] + [n/k^2] + [n/k^3] + [n/k^4] + ... for n, k > 1.

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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.

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