A146291 -id:A146291 - cp's OEIS frontend

A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

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

Offset: 1

Matthew Vandermast, Nov 11 2008

The formula used in obtaining the A025487(n)th row (see below) also gives the number of divisors of the k-th power of A025487(n).
Every row that appears in A146289 appears exactly once in the table. Rows appear in order of first appearance in A146289.
T(n,0)=1.

			Rows begin:
  1;
  1,1;
  1,2;
  1,2,1;
  1,3;
  1,3,2;
  1,4;
  1,4,3;...
36's 9 divisors include 1 divisor with 0 distinct prime factors (1); 4 with 1 (2, 3, 4 and 9); and 4 with 2 (6, 12, 18 and 36). Since 36 = A025487(11), the 11th row of the table therefore reads (1, 4, 4). These are the positive coefficients of the polynomial equation 1 + 4k + 4k^2 = (1 + 2k)(1 + 2k), derived from the prime factorization of 36 (namely, 2^2*3^2).

Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Distinct Prime Factors
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

For the number of distinct prime factors of n, see A001221.
Row sums equal A146288(n). T(n, 1)=A036041(n) for n>1. T(n, A061394(n))=A052306(n).
Row A098719(n) of this table is identical to row n of A007318.
Cf. A146289. Also cf. A146291, A146292.

If A025487(n)'s canonical factorization into prime powers is Product p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (1 + ek).

A181567 Triangle read by rows: T(n,k) is coefficient of k-th power in expansion of ((x^(n+1)-1)/(x-1))^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10, 20, 35, 52, 68, 80, 85, 80, 68, 52, 35, 20, 10, 4, 1, 1, 5, 15, 35, 70, 126, 205, 305, 420, 540, 651, 735, 780, 780, 735, 651, 540, 420, 305, 205, 126, 70, 35, 15, 5, 1, 1, 6, 21, 56, 126, 252, 462, 786, 1251

Offset: 0

Matthew Vandermast, Oct 31 2010

In each row n>=0, k takes values from 0 to n^2 inclusive. Row sums equal A000169(n+1). All rows are palindromic. Row n is also row n of the (n+1)-nomial array (e.g., row 1 is also row 1 of A007318).
T(n,k) gives the number of divisors of A181555(n) with k prime factors counted with multiplicity. See also A001222, A071207, A146291, A146292.
T(n,k) is the number of size k submultisets of the so-called regular multiset {1_1,1_2,...,1_(n-1),1_n, ... ,i_1,i_2,...,i_(n-1),i_n, ... ,n_1,n_2,...,n_(n-1),n_n} (which contains n copies of i for 0 < i < n). - Thomas Wieder, Dec 28 2013

			Rows begin:
1;
1,1;
1,2,3,2,1;
1,3,6,10,12,12,10,6,3,1;...
T(n=3,k=4) = 12 because we have 12 submultisets (without regard of the order of elements) of size k=4 for the regular multiset (n=3) {1, 1, 1, 2, 2, 2, 3, 3, 3}: {1, 1, 1, 2}, {1, 1, 1, 3}, {1, 1, 2, 2}, {1, 1, 2, 3}, {1, 1, 3, 3}, {1, 2, 2, 2}, {1, 2, 2, 3}, {1, 2, 3, 3}, {1, 3, 3, 3}, {2, 2, 2, 3}, {2, 2, 3, 3}, {2, 3, 3, 3}.

Alois P. Heinz, Rows n = 0..32, flattened
Anonymous, Polynomial calculator
Thomas Wieder, A181567 as Excel table
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

A163181 gives row n of n-nomial array. See also A000012, A007318, A027907, A008287, A035343, A063260, A063265, A171890.

b:= proc(n, k, i) option remember; `if`(k=0, 1,
     `if`(i<1, 0, add(b(n, k-j, i-1), j=0..n)))
    end:
T:= (n, k)-> b(n, k, n):
seq(seq(T(n, k), k=0..n^2), n=0..8); # Alois P. Heinz, Jul 04 2016

row[n_] := CoefficientList[((x^(n+1) - 1)/(x-1))^n + O[x]^(n^2+1), x]; Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Apr 06 2017 *)

A343662 Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 4, 5, 2, 1, 2, 1, 1, 4, 6, 4, 1, 1, 3, 3, 1, 1, 4, 5, 2, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 4, 5, 2, 1, 4, 5, 2, 1, 5, 10, 10, 5, 1, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 6, 12, 10, 3, 1, 4, 5, 2, 1, 4, 5, 2

Offset: 1

Gus Wiseman, May 01 2021

			Triangle begins:
   1:  1  1
   2:  1  2  1
   3:  1  2  1
   4:  1  3  3  1
   5:  1  2  1
   6:  1  4  5  2
   7:  1  2  1
   8:  1  4  6  4  1
   9:  1  3  3  1
  10:  1  4  5  2
  11:  1  2  1
  12:  1  6 12 10  3
  13:  1  2  1
  14:  1  4  5  2
  15:  1  4  5  2
  16:  1  5 10 10  5  1
For example, row n = 12 counts the following chains:
  ()  (1)   (2/1)   (4/2/1)   (12/4/2/1)
      (2)   (3/1)   (6/2/1)   (12/6/2/1)
      (3)   (4/1)   (6/3/1)   (12/6/3/1)
      (4)   (4/2)   (12/2/1)
      (6)   (6/1)   (12/3/1)
      (12)  (6/2)   (12/4/1)
            (6/3)   (12/4/2)
            (12/1)  (12/6/1)
            (12/2)  (12/6/2)
            (12/3)  (12/6/3)
            (12/4)
            (12/6)

Column k = 1 is A000005.
Row ends are A008480.
Row lengths are A073093.
Column k = 2 is A238952.
The case from n to 1 is A334996 or A251683 (row sums: A074206).
A non-strict version is A334997 (transpose: A077592).
The case starting with n is A337255 (row sums: A067824).
Row sums are A337256 (nonempty: A253249).
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A097805 counts compositions by sum and length.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A163767 counts length n - 1 chains of divisors of n.
A167865 counts strict chains of divisors > 1 summing to n.
A337070 counts strict chains of divisors starting with superprimorials.
Cf. A002033, A007425, A007426, A051026, A062319, A143773, A186972, A327527, A337074, A337105, A337107, A343658.

Table[Length[Select[Reverse/@Subsets[Divisors[n],{k}],And@@Divisible@@@Partition[#,2,1]&]],{n,15},{k,0,PrimeOmega[n]+1}]

A146292 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A036041(n)), giving the number of divisors of A025487(n) with m prime factors (counted with multiplicity).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 3, 4, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 3, 2, 1, 1, 3, 5, 5, 3, 1, 1, 2, 2, 2, 2

Offset: 1

Matthew Vandermast, Nov 11 2008

All rows are palindromic. T(n, 0) = T(n, A036041(n)) = 1.

Every row that appears in A146291 appears exactly once in the table. Rows appear in order of first appearance in A146291.

			Rows begin:
  1;
  1,1;
  1,1,1;
  1,2,1;
  1,1,1,1;
  1,2,2,1;
  1,1,1,1,1;...
36's 9 divisors include 1 divisor with 0 total prime factors (1);, 2 with 1 (2 and 3); 3 with 2 (4, 6 and 9); 2 with 3 (12 and 18); and 1 with 4 (36). Since 36 = A025487(11), the 11th row of the table therefore reads (1, 2, 3, 2, 1). These are the positive coefficients of the polynomial 1 + 2k + 3k^2 + 2k^3 + (1)k^4 = (1 + k + k^2)(1 + k + k^2), derived from the prime factorization of 36 (namely, 2^2*3^2).

Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Roundness
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

For the number of prime factors of n counted with multiplicity, see A001222.
Row sums equal A146288(n). T(n, 1) = A061394(n) for n>1.
Row A098719(n) of this table is identical to row n of A007318.
Cf. A146291. Also cf. A146289, A146290.

If A025487(n)'s canonical factorization into prime powers is the product of p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (sum_{i=0..e} k^i).

A336498 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with k prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 4, 4, 3, 1, 1, 3, 5, 6, 6, 5, 3, 1, 1, 4, 8, 11, 12, 11, 8, 4, 1, 1, 4, 8, 11, 12, 12, 12, 12, 11, 8, 4, 1, 1, 4, 8, 12, 16, 19, 20, 20, 19, 16, 12, 8, 4, 1, 1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1

Offset: 0

Gus Wiseman, Aug 03 2020

Row n is row n! of A146291. Row lengths are A022559(n) + 1.

			Triangle begins:
  1
  1
  1  1
  1  2  1
  1  2  2  2  1
  1  3  4  4  3  1
  1  3  5  6  6  5  3  1
  1  4  8 11 12 11  8  4  1
  1  4  8 11 12 12 12 12 11  8  4  1
  1  4  8 12 16 19 20 20 19 16 12  8  4  1
Row n = 6 counts the following divisors:
  1  2   4   8  16   48  144  720
     3   6  12  24   72  240
     5   9  18  36   80  360
        10  20  40  120
        15  30  60  180
            45  90
Row n = 7 counts the following divisors:
  1  2   4    8   16   48   144   720  5040
     3   6   12   24   72   240  1008
     5   9   18   36   80   336  1680
     7  10   20   40  112   360  2520
        14   28   56  120   504
        15   30   60  168   560
        21   42   84  180   840
        35   45   90  252  1260
             63  126  280
             70  140  420
            105  210  630
                 315

A000720 is column k = 1.
A008302 is the version for superprimorials.
A022559 gives row lengths minus one.
A027423 gives row sums.
A146291 is the generalization to non-factorials.
A336499 is the restriction to divisors in A130091.
A000142 lists factorial numbers.
A336415 counts uniform divisors of n!.
Cf. A000005, A001222, A118914, A124010, A181796, A327526, A336420.
Factorial numbers: A002982, A007489, A048656, A054991, A071626, A325272, A325617, A336414, A336415, A336416, A336418.

Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&]],{n,0,10},{k,0,PrimeOmega[n!]}]

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

Gus Wiseman, Apr 29 2021

nonn

			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

Chai Wah Wu, Table of n, a(n) for n = 1..10000

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.

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

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

A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 0, 1, 3, 2, 5, 3, 3, 2, 1, 1, 4, 2, 7, 4, 4, 3, 2, 0, 1, 4, 2, 7, 4, 5, 7, 7, 6, 3, 2, 0, 1, 4, 2, 8, 8, 9, 10, 11, 11, 7, 8, 5, 2, 0, 1, 4, 3, 11, 8, 11, 16, 16, 15, 15, 15, 13, 9, 6, 3, 1, 1, 5, 3, 14, 10, 13, 21, 21, 20, 19, 21, 18, 13, 9, 5, 2, 0

Offset: 0

Gus Wiseman, Aug 03 2020

Row lengths are A022559(n) + 1.

			Triangle begins:
  1
  1
  1  1
  1  2  0
  1  2  1  2  1
  1  3  1  3  2  0
  1  3  2  5  3  3  2  1
  1  4  2  7  4  4  3  2  0
  1  4  2  7  4  5  7  7  6  3  2  0
  1  4  2  8  8  9 10 11 11  7  8  5  2  0
  1  4  3 11  8 11 16 16 15 15 15 13  9  6  3  1
  1  5  3 14 10 13 21 21 20 19 21 18 13  9  5  2  0
  1  5  3 14 10 14 25 23 27 24 30 28 28 25 20 16 11  5  2  0
Row n = 7 counts the following divisors:
  1  2  4  8   16  48   144  720   {}
     3  9  12  24  72   360  1008
     5     18  40  80   504
     7     20  56  112
           28
           45
           63

A000720 is column k = 1.
A022559 gives row lengths minus one.
A056172 appears to be column k = 2.
A336414 gives row sums.
A336420 is the version for superprimorials.
A336498 is the version counting all divisors.
A336865 is the generalization to non-factorials.
A336866 lists indices of rows with a final 1.
A336867 lists indices of rows with a final 0.
A336868 gives the final terms in each row.
A000110 counts divisors of superprimorials with distinct prime exponents.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
Cf. A000005, A001222, A008278, A098859, A118914, A124010, A146291, A336422, A336500.
Factorial numbers: A000142, A002982, A027423, A048656, A048742, A054991, A071626, A336425, A336617.

Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,6},{k,0,PrimeOmega[n!]}]

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

Gus Wiseman, May 05 2021

nonn

			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}

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.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n],{n,25}]

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

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

A336865 Irregular triangle read by rows where T(n,k) is the number of divisors of n with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 0, 0

Offset: 1

Gus Wiseman, Aug 06 2020

Row lengths are A073093(n) = A001222(n) + 1.

			The triangle begins as follows. The n-th row is shown to the right of "n:".
     1: (1)          16: (1,1,1,1,1)    31: (1,1)
     2: (1,1)        17: (1,1)          32: (1,1,1,1,1,1)
     3: (1,1)        18: (1,2,1,1)      33: (1,2,0)
     4: (1,1,1)      19: (1,1)          34: (1,2,0)
     5: (1,1)        20: (1,2,1,1)      35: (1,2,0)
     6: (1,2,0)      21: (1,2,0)        36: (1,2,2,2,0)
     7: (1,1)        22: (1,2,0)        37: (1,1)
     8: (1,1,1,1)    23: (1,1)          38: (1,2,0)
     9: (1,1,1)      24: (1,2,1,2,1)    39: (1,2,0)
    10: (1,2,0)      25: (1,1,1)        40: (1,2,1,2,1)
    11: (1,1)        26: (1,2,0)        41: (1,1)
    12: (1,2,1,1)    27: (1,1,1,1)      42: (1,3,0,0)
    13: (1,1)        28: (1,2,1,1)      43: (1,1)
    14: (1,2,0)      29: (1,1)          44: (1,2,1,1)
    15: (1,2,0)      30: (1,3,0,0)      45: (1,2,1,1)
Row n = 72 counts the following divisors:
  1  2  4   8  24  72
     3  9  12
           18
Row n = 1200 counts the following divisors:
  1  2   4   8  16   48  400  1200
     3  25  12  24   80  600
     5      20  40  200
            50
            75

A073093 gives row lengths.
A130092 gives positions of rows ending with 0.
A146291 is the version not requiring distinct prime multiplicities.
A181796 gives row sums.
A336499 is the restriction to factorial numbers.
A001222 counts prime factors, counting multiplicity.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor of n with distinct prime multiplicities.
A336423 counts chains using A130091.
Cf. A000005, A001221, A098859, A118914, A124010, A336422, A336500, A336571.

Table[Length[Select[Divisors[n],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,20},{k,0,PrimeOmega[n]}]

A343661 Sum of numbers of y-multisets of divisors of x for each x >= 1, y >= 0, x + y = n.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 46, 70, 105, 155, 223, 316, 443, 619, 865, 1210, 1690, 2354, 3263, 4497, 6157, 8368, 11280, 15078, 19989, 26296, 34356, 44626, 57693, 74321, 95503, 122535, 157101, 201377, 258155, 330994, 424398, 544035, 696995, 892104, 1140298, 1455080

Offset: 1

Gus Wiseman, Apr 30 2021

nonn

			The a(5) = 12 multisets of divisors:
  {1,1,1,1}  {1,1,1}  {1,1}  {1}  {}
             {1,1,2}  {1,3}  {2}
             {1,2,2}  {3,3}  {4}
             {2,2,2}

Antidiagonal sums of the array A343658 (or row sums of the triangle).
Dominates A343657.
A000005 counts divisors.
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
A343656 counts divisors of powers.
Cf. A000169, A000312, A009998, A062319, A067824, A143773, A146291, A176029, A184389, A285572, A326077, A327527, A334996, A343652, A343657.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[Sum[multchoo[DivisorSigma[0,k],n-k],{k,n}],{n,10}]

a(n) = Sum_{k=1..n} binomial(sigma(k) + n - k - 1, n - k).

cp's OEIS Frontend

A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

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A181567 Triangle read by rows: T(n,k) is coefficient of k-th power in expansion of ((x^(n+1)-1)/(x-1))^n.

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A343662 Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.

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A146292 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A036041(n)), giving the number of divisors of A025487(n) with m prime factors (counted with multiplicity).

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A336498 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with k prime factors, counted with multiplicity.

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

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A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

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

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A336865 Irregular triangle read by rows where T(n,k) is the number of divisors of n with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

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