A327527 -id:A327527 - cp's OEIS frontend

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

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).

A343939 Number of n-chains of divisors of n.

Original entry on oeis.org

1, 3, 4, 15, 6, 49, 8, 165, 55, 121, 12, 1183, 14, 225, 256, 4845, 18, 3610, 20, 4851, 484, 529, 24, 73125, 351, 729, 4060, 12615, 30, 29791, 32, 435897, 1156, 1225, 1296, 494209, 38, 1521, 1600, 505981, 42, 79507, 44, 46575, 49726, 2209, 48

Offset: 1

Gus Wiseman, May 05 2021

nonn

			The a(1) = 1 through a(5) = 6 chains:
  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)
       (2/1)  (3/1/1)  (2/1/1/1)  (5/1/1/1/1)
       (2/2)  (3/3/1)  (2/2/1/1)  (5/5/1/1/1)
              (3/3/3)  (2/2/2/1)  (5/5/5/1/1)
                       (2/2/2/2)  (5/5/5/5/1)
                       (4/1/1/1)  (5/5/5/5/5)
                       (4/2/1/1)
                       (4/2/2/1)
                       (4/2/2/2)
                       (4/4/1/1)
                       (4/4/2/1)
                       (4/4/2/2)
                       (4/4/4/1)
                       (4/4/4/2)
                       (4/4/4/4)

Diagonal n = k - 1 of the array A077592.
Chains of length n - 1 are counted by A163767.
Diagonal n = k of the array A334997.
The version counting all multisets of divisors (not just chains) is A343935.
A000005(n) counts divisors of n.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
A251683(n,k-1) counts strict k-chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict k-chains of divisors from n to 1.
A337255(n,k) counts strict k-chains of divisors starting with n.
A343658(n,k) counts k-multisets of divisors of n.
A343662(n,k) counts strict k-chains of divisors of n (row sums: A337256).
Cf. A062319, A066959, A143773, A176029, A327527, A343656.

Table[Length[Select[Tuples[Divisors[n],n],OrderedQ[#]&&And@@Divisible@@@Reverse/@Partition[#,2,1]&]],{n,10}]

A368251 The number of nonsquarefree divisors of n that are powers of squarefree numbers (A072777).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 5, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Dec 19 2023

nonn

First differs from A046660 and A066301 at n = 36, and from A183094 at n = 72.

Let b(n, k) be the sequence that counts the divisors of n that are k-th powers of squarefree numbers. Then, b(n, 1) = A034444(n), b(n, 2) = A323308(n), b(n, 3) = A368248(n). b(n, k) is multiplicative with b(p^e, k) = 2 if e >= k, and 1 otherwise. The asymptotic mean of b(n, k) for k >= 2 is lim_{m->oo} (1/m) * Sum_{n=1..m} b(n, k) = zeta(k)/zeta(2*k). Since a(n) = Sum_{k>=2} (b(n, k) - 1), the formula for the asymptotic mean of this sequence follows (see the Formula section).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005117, A048105, A072777, A327527, A368250.
Cf. A034444, A323308, A368248.
Cf. A046660, A066301, A183094.

a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1] - 2^Length[e]]; a[1] = 0; Array[a, 100]

a(n) = {my(f = factor(n), e, m, h, c); if(n == 1, 0, e = f[,2]; m = vecmax(e); h = vector(m); for(i = 1,m, c = 0; for(j = 1, #e, if(e[j] == (m+1-i), c++)); h[i] = c); for(i = 2, m, h[i] += h[i-1]); for(i = 1, m, h[i] = 2^h[i]-1); 1 + vecsum(h) - 1<<#e);}

a(n) = A327527(n) - A034444(n).
a(n) = 0 if and only if n is squarefree (A005117).
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=2} (zeta(k)/zeta(2*k) - 1) = 0.848633... (A368250).

A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 66, 95, 135, 187, 256, 346, 463, 613, 803, 1040, 1336, 1703, 2158, 2720, 3409, 4244, 5251, 6461, 7911, 9643, 11707, 14157, 17058, 20480, 24502, 29212, 34707, 41094, 48496, 57053, 66926, 78296, 91369, 106376, 123581, 143276, 165786

Offset: 1

Gus Wiseman, May 07 2021

nonn

			The a(8) = 45 chains:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)  (1/1/1/1/1/1)
      (7)  (2/1)  (5/1/1)  (2/1/1/1)  (3/1/1/1/1)  (2/1/1/1/1/1)
           (2/2)  (5/5/1)  (2/2/1/1)  (3/3/1/1/1)  (2/2/1/1/1/1)
           (3/1)  (5/5/5)  (2/2/2/1)  (3/3/3/1/1)  (2/2/2/1/1/1)
           (3/3)           (2/2/2/2)  (3/3/3/3/1)  (2/2/2/2/1/1)
           (6/1)           (4/1/1/1)  (3/3/3/3/3)  (2/2/2/2/2/1)
           (6/2)           (4/2/1/1)               (2/2/2/2/2/2)
           (6/3)           (4/2/2/1)
           (6/6)           (4/2/2/2)
                           (4/4/1/1)
                           (4/4/2/1)           (1/1/1/1/1/1/1)
                           (4/4/2/2)
                           (4/4/4/1)
                           (4/4/4/2)
                           (4/4/4/4)

Antidiagonal sums of the array (or row sums of the triangle) A334997.
A000005 counts divisors of n.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1.
A146291 counts divisors of n with k prime factors (with multiplicity).
A251683 counts strict length k + 1 chains of divisors from n to 1.
A253249 counts nonempty chains of divisors of n.
A334996 counts strict length k chains of divisors from n to 1.
A337255 counts strict length k chains of divisors starting with n.
Array version of A334997 has:
- column k = 2 A007425,
- transpose A077592,
- subdiagonal n = k + 1 A163767,
- strict case A343662 (row sums: A337256),
- version counting all multisets of divisors (not just chains) A343658,
- diagonal n = k A343939.
Cf. A018892, A062319, A066959, A143773, A176029, A327527, A343656.

Total/@Table[Length[Select[Tuples[Divisors[n-k],k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,0,n-1}]

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

Gus Wiseman, May 05 2021

nonn

			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

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.

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

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

A351394 Number of divisors of n that are either squarefree, prime powers, or both.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 8, 2, 6, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 9, 2, 4, 5, 7, 4, 8, 2, 5, 4, 8, 2, 7, 2, 4, 5, 5, 4, 8, 2, 7, 5, 4, 2, 9, 4, 4, 4, 6, 2, 9, 4, 5, 4, 4, 4

Offset: 1

Wesley Ivan Hurt, Feb 09 2022

			a(36) = 6; 36 has 4 squarefree divisors 1,2,3,6 (where the primes 2 and 3 are both squarefree and 1st powers of primes) and 2 (additional) divisors that are powers of primes, 2^2 and 3^2.

Cf. A001221, A008683, A046660, A048105, A246655.

Cf. Similar to A327527.

a[n_] := Module[{e = FactorInteger[n][[;;, 2]], nu, omega}, nu = Length[e]; omega = Total[e]; 2^nu + omega - nu]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Oct 06 2023 *)

a(n) = {my(f = factor(n), nu = omega(f), om = bigomega(f)); 2^nu + om - nu;} \\ Amiram Eldar, Oct 06 2023

a(n) = Sum_{d|n} sign(mu(d)^2 + [omega(d) = 1]).
a(n) = Sum_{d|n} (mu(d)^2 + [omega(d) = 1]*(1 - mu(d)^2)).
a(n) = A048105(n) + A046660(n). - Amiram Eldar, Oct 06 2023

A336871 Number of divisors d of A076954(n) with distinct prime multiplicities such that the numerator of A006939(n)/d also has distinct prime multiplicities.

Original entry on oeis.org

1, 2, 4, 11, 28, 96, 309, 1256, 4676, 21647

Offset: 0

Gus Wiseman, Aug 06 2020

The sequence A006939 is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

The sequence A076954 is A076954(n) = Product_{i=1..n} prime(i)^i.

			The a(0) = 1 through a(3) = 11 divisors:
  1  2  18   2250
     1   9   1125
         3    375
         1    125
               75
               45
               25
               18
                9
                5
                1

A336419 is the version for superprimorials.
A336500 is the generalization to all positive integers.
A000005 counts divisors.
A006939 lists superprimorials or Chernoff numbers.
A007425 counts divisors of divisors.
A076954 is a sister of superprimorials.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327523 counts factorizations of elements of A130091 using elements of A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
Cf. A001055, A022559, A022915, A027423, A091050, A124010, A317829, A327498, A327527, A336420, A336421, A336571.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
cochern[n_]:=Product[Prime[i]^i,{i,n}];
Table[Length[Select[Divisors[cochern[n]],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[chern[n]/#]&]],{n,0,5}]

cp's OEIS Frontend

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

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A343939 Number of n-chains of divisors of n.

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A368251 The number of nonsquarefree divisors of n that are powers of squarefree numbers (A072777).

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A343940 Sum of numbers of ways to choose a k-chain of divisors of n - k, for k = 0..n - 1.

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

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A351394 Number of divisors of n that are either squarefree, prime powers, or both.

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A336871 Number of divisors d of A076954(n) with distinct prime multiplicities such that the numerator of A006939(n)/d also has distinct prime multiplicities.

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