cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A343658 Array read by antidiagonals where A(n,k) is the number of ways to choose a multiset of k divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 10, 2, 1, 1, 8, 7, 21, 5, 20, 3, 4, 1, 1, 9, 8, 28, 6, 35, 4, 10, 3, 1, 1, 10, 9, 36, 7, 56, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 84, 6, 35, 10, 10, 2, 1
Offset: 1

Views

Author

Gus Wiseman, Apr 29 2021

Keywords

Comments

First differs from A343656 at A(4,2) = 6, A343656(4,2) = 5.
As a triangle, T(n,k) = number of ways to choose a multiset of n - k divisors of k.

Examples

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=1:  1   1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8   9
  n=3:  1   2   3   4   5   6   7   8   9
  n=4:  1   3   6  10  15  21  28  36  45
  n=5:  1   2   3   4   5   6   7   8   9
  n=6:  1   4  10  20  35  56  84 120 165
  n=7:  1   2   3   4   5   6   7   8   9
  n=8:  1   4  10  20  35  56  84 120 165
  n=9:  1   3   6  10  15  21  28  36  45
Triangle begins:
   1
   1   1
   1   2   1
   1   3   2   1
   1   4   3   3   1
   1   5   4   6   2   1
   1   6   5  10   3   4   1
   1   7   6  15   4  10   2   1
   1   8   7  21   5  20   3   4   1
   1   9   8  28   6  35   4  10   3   1
   1  10   9  36   7  56   5  20   6   4   1
   1  11  10  45   8  84   6  35  10  10   2   1
For example, row n = 6 counts the following multisets:
  {1,1,1,1,1}  {1,1,1,1}  {1,1,1}  {1,1}  {1}  {}
               {1,1,1,2}  {1,1,3}  {1,2}  {5}
               {1,1,2,2}  {1,3,3}  {1,4}
               {1,2,2,2}  {3,3,3}  {2,2}
               {2,2,2,2}           {2,4}
                                   {4,4}
Note that for n = 6, k = 4 in the triangle, the two multisets {1,4} and {2,2} represent the same divisor 4, so they are only counted once under A343656(4,2) = 5.

Links

Crossrefs

Row k = 1 of the array is A000005.
Column n = 4 of the array is A000217.
Column n = 6 of the array is A000292.
Row k = 2 of the array is A184389.
The distinct products of these multisets are counted by A343656.
Antidiagonal sums of the array (or row sums of the triangle) are A343661.
A000312 = n^n.
A009998(n,k) = n^k (as an array, offset 1).
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
Cf. A000169, A062319, A067824, A143773, A146291, A176029, A285572, A326077, A327527, A334996, A343652, A343657.

Programs

  • Mathematica
    multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,k],n-k],{n,10},{k,n}]
  • PARI
    A(n,k) = binomial(numdiv(n) + k - 1, k)
{ for(n=1, 9, for(k=0, 8, print1(A(n,k), ", ")); print ) } \\ Andrew Howroyd, Jan 11 2024

Formula

A(n,k) = ((A000005(n), k)) = A007318(A000005(n) + k - 1, k).
T(n,k) = ((A000005(k), n - k)) = A007318(A000005(k) + n - k - 1, n - k).

A343657 Sum of number of divisors of x^y for each x >= 1, y >= 0, x + y = n.

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 27, 39, 56, 77, 103, 134, 174, 223, 283, 356, 445, 547, 666, 802, 959, 1139, 1344, 1574, 1835, 2128, 2454, 2815, 3213, 3648, 4126, 4653, 5239, 5888, 6608, 7407, 8298, 9288, 10385, 11597, 12936, 14408, 16025, 17799, 19746, 21882, 24221
Offset: 1

Views

Author

Gus Wiseman, Apr 29 2021

Keywords

Examples

			The a(7) = 27 divisors:
  1  32  81  64  25  6  1
     16  27  32  5   3
     8   9   16  1   2
     4   3   8       1
     2   1   4
     1       2
             1

Links

Crossrefs

Antidiagonal row sums (row sums of the triangle) of A343656.
Dominated by A343661.
A000005(n) counts divisors of n.
A000312(n) = n^n.
A007318(n,k) counts k-sets of elements of {1..n}.
A009998(n,k) = n^k (as an array, offset 1).
A059481(n,k) counts k-multisets of elements of {1..n}.
A343658(n,k) counts k-multisets of divisors of n.
Cf. A000169, A000272, A002064, A002109, A048691, A062319, A066959, A143773, A146291, A176029, A251683, A282935, A326358, A327527, A334996.

Programs

  • Mathematica
    Total/@Table[DivisorSigma[0,k^(n-k)],{n,30},{k,n}]

Formula

a(n) = Sum_{k=1..n} A000005(k^(n-k)).

A343935 Number of ways to choose a multiset of n divisors of n.

Original entry on oeis.org

1, 3, 4, 15, 6, 84, 8, 165, 55, 286, 12, 6188, 14, 680, 816, 4845, 18, 33649, 20, 53130, 2024, 2300, 24, 2629575, 351, 3654, 4060, 237336, 30, 10295472, 32, 435897, 7140, 7770, 8436, 177232627, 38, 10660, 11480, 62891499, 42, 85900584, 44, 1906884, 2118760
Offset: 1

Views

Author

Gus Wiseman, May 05 2021

Keywords

Examples

			The a(1) = 1 through a(5) = 6 multisets:
  {1}  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}
       {1,2}  {1,1,3}  {1,1,1,2}  {1,1,1,1,5}
       {2,2}  {1,3,3}  {1,1,1,4}  {1,1,1,5,5}
              {3,3,3}  {1,1,2,2}  {1,1,5,5,5}
                       {1,1,2,4}  {1,5,5,5,5}
                       {1,1,4,4}  {5,5,5,5,5}
                       {1,2,2,2}
                       {1,2,2,4}
                       {1,2,4,4}
                       {1,4,4,4}
                       {2,2,2,2}
                       {2,2,2,4}
                       {2,2,4,4}
                       {2,4,4,4}
                       {4,4,4,4}

Crossrefs

Diagonal n = k of A343658.
Choosing n divisors of n - 1 gives A343936.
The version for chains of divisors is A343939.
A000005 counts divisors.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998 = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
A146291 counts divisors of n with k prime factors (with multiplicity).
A253249 counts nonempty chains of divisors of n.
Strict chains of divisors:
- A067824 counts strict chains of divisors starting with n.
- A074206 counts strict chains of divisors from n to 1.
- A251683 counts strict length k + 1 chains of divisors from n to 1.
- A334996 counts strict length-k chains of divisors from n to 1.
- A337255 counts strict length-k chains of divisors starting with n.
- A337256 counts strict chains of divisors of n.
- A343662 counts strict length-k chains of divisors.
Cf. A062319, A143773, A163767, A176029, A327527, A334997, A343661.

Programs

  • Mathematica
    multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n],{n,25}]
  • Python
    from math import comb
from sympy import divisor_count
def A343935(n): return comb(divisor_count(n)+n-1,n) # Chai Wah Wu, Jul 05 2024

Formula

a(n) = ((sigma(n), n)) = binomial(sigma(n) + n - 1, n) where sigma = A000005 and binomial = A007318.

A343936 Number of ways to choose a multiset of n divisors of n - 1.

Original entry on oeis.org

1, 2, 3, 10, 5, 56, 7, 120, 45, 220, 11, 4368, 13, 560, 680, 3876, 17, 26334, 19, 42504, 1771, 2024, 23, 2035800, 325, 3276, 3654, 201376, 29, 8347680, 31, 376992, 6545, 7140, 7770, 145008513, 37, 9880, 10660, 53524680, 41, 73629072, 43, 1712304, 1906884
Offset: 1

Views

Author

Gus Wiseman, May 05 2021

Keywords

Examples

			The a(1) = 1 through a(5) = 5 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
      {2}  {1,3}  {1,1,2}  {1,1,1,5}
           {3,3}  {1,1,4}  {1,1,5,5}
                  {1,2,2}  {1,5,5,5}
                  {1,2,4}  {5,5,5,5}
                  {1,4,4}
                  {2,2,2}
                  {2,2,4}
                  {2,4,4}
                  {4,4,4}
The a(6) = 56 multisets:
  11111  11136  11333  12236  13366  22266  23666
  11112  11166  11336  12266  13666  22333  26666
  11113  11222  11366  12333  16666  22336  33333
  11116  11223  11666  12336  22222  22366  33336
  11122  11226  12222  12366  22223  22666  33366
  11123  11233  12223  12666  22226  23333  33666
  11126  11236  12226  13333  22233  23336  36666
  11133  11266  12233  13336  22236  23366  66666

Crossrefs

The version for chains of divisors is A163767.
Diagonal n = k + 1 of A343658.
Choosing n divisors of n gives A343935.
A000005 counts divisors.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998 = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
A146291 counts divisors of n with k prime factors (with multiplicity).
A253249 counts nonempty chains of divisors of n.
Strict chains of divisors:
- A067824 counts strict chains of divisors starting with n.
- A074206 counts strict chains of divisors from n to 1.
- A251683 counts strict length k + 1 chains of divisors from n to 1.
- A334996 counts strict length-k chains of divisors from n to 1.
- A337255 counts strict length-k chains of divisors starting with n.
- A337256 counts strict chains of divisors of n.
- A343662 counts strict length-k chains of divisors.
Cf. A062319, A143773, A163767, A176029, A327527, A334997, A343656, A343661.

Programs

  • Mathematica
    multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n-1],{n,50}]

Formula

a(n) = ((sigma(n - 1), n)) = binomial(sigma(n - 1) + n - 1, n) where sigma = A000005 and binomial = A007318.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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