A343656 -id:A343656 - cp's OEIS frontend

A343526 Number of divisors of n^7.

1, 8, 8, 15, 8, 64, 8, 22, 15, 64, 8, 120, 8, 64, 64, 29, 8, 120, 8, 120, 64, 64, 8, 176, 15, 64, 22, 120, 8, 512, 8, 36, 64, 64, 64, 225, 8, 64, 64, 176, 8, 512, 8, 120, 120, 64, 8, 232, 15, 120, 64, 120, 8, 176, 64, 176, 64, 64, 8, 960, 8, 64, 120, 43, 64, 512, 8, 120, 64, 512, 8

Offset: 1

Seiichi Manyama, May 15 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=7 of A343656.

Cf. A000005, A001015.

Table[DivisorSigma[0, n^7], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = numdiv(n^7);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 7*f[k]+1);

PARI
```
a(n) = sumdiv(n, d, 7^omega(d));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 7^omega(k)*x^k/(1-x^k)))

for(n=1, 100, print1(direuler(p=2, n, (1 + 6*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

a(n) = A000005(A001015(n)).
Multiplicative with a(p^e) = 7*e+1.
a(n) = Sum_{d|n} 7^omega(d).
G.f.: Sum_{k>=1} 7^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 6/p^s). - Vaclav Kotesovec, Aug 19 2021

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}]

A344327 Number of divisors of n^4.

Original entry on oeis.org

1, 5, 5, 9, 5, 25, 5, 13, 9, 25, 5, 45, 5, 25, 25, 17, 5, 45, 5, 45, 25, 25, 5, 65, 9, 25, 13, 45, 5, 125, 5, 21, 25, 25, 25, 81, 5, 25, 25, 65, 5, 125, 5, 45, 45, 25, 5, 85, 9, 45, 25, 45, 5, 65, 25, 65, 25, 25, 5, 225, 5, 25, 45, 25, 25, 125, 5, 45, 25, 125, 5, 117, 5, 25, 45, 45, 25, 125, 5, 85, 17, 25

Offset: 1

Seiichi Manyama, May 15 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=4 of A343656.

Cf. A000005, A000583, A202994.

Table[DivisorSigma[0, n^4], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = numdiv(n^4);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 4*f[k]+1);

PARI
```
a(n) = sumdiv(n, d, 4^omega(d));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 4^omega(k)*x^k/(1-x^k)))

for(n=1, 100, print1(direuler(p=2, n, (1 + 3*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, May 15 2021

for(n=1, 100, print1(direuler(p=2, n, (1 - 6*X^2 + 8*X^3 - 3*X^4)/(1 - X)^5)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

a(n) = A000005(A000583(n)).
Multiplicative with a(p^e) = 4*e+1.
a(n) = Sum_{d|n} 4^omega(d).
G.f.: Sum_{k>=1} 4^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 3/p^s). - Vaclav Kotesovec, May 15 2021
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 6/p^(2*s) + 8/p^(3*s) - 3/p^(4*s)). - Vaclav Kotesovec, Aug 20 2021

A344328 Number of divisors of n^5.

Original entry on oeis.org

1, 6, 6, 11, 6, 36, 6, 16, 11, 36, 6, 66, 6, 36, 36, 21, 6, 66, 6, 66, 36, 36, 6, 96, 11, 36, 16, 66, 6, 216, 6, 26, 36, 36, 36, 121, 6, 36, 36, 96, 6, 216, 6, 66, 66, 36, 6, 126, 11, 66, 36, 66, 6, 96, 36, 96, 36, 36, 6, 396, 6, 36, 66, 31, 36, 216, 6, 66, 36, 216, 6, 176, 6, 36, 66, 66, 36

Offset: 1

Seiichi Manyama, May 15 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=5 of A343656.

Cf. A000005, A000584, A082476 (5^omega(n)), A203556.

Table[DivisorSigma[0, n^5], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = numdiv(n^5);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 5*f[k]+1);

PARI
```
a(n) = sumdiv(n, d, 5^omega(d));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 5^omega(k)*x^k/(1-x^k)))

for(n=1, 100, print1(direuler(p=2, n, (1 + 4*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

a(n) = A000005(A000584(n)).
Multiplicative with a(p^e) = 5*e+1.
a(n) = Sum_{d|n} 5^omega(d).
G.f.: Sum_{k>=1} 5^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 4/p^s). - Vaclav Kotesovec, Aug 19 2021

A344329 Number of divisors of n^6.

Original entry on oeis.org

1, 7, 7, 13, 7, 49, 7, 19, 13, 49, 7, 91, 7, 49, 49, 25, 7, 91, 7, 91, 49, 49, 7, 133, 13, 49, 19, 91, 7, 343, 7, 31, 49, 49, 49, 169, 7, 49, 49, 133, 7, 343, 7, 91, 91, 49, 7, 175, 13, 91, 49, 91, 7, 133, 49, 133, 49, 49, 7, 637, 7, 49, 91, 37, 49, 343, 7, 91, 49, 343, 7, 247, 7

Offset: 1

Seiichi Manyama, May 15 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=6 of A343656.

Cf. A000005, A001014.

Table[DivisorSigma[0, n^6], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = numdiv(n^6);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 6*f[k]+1);

PARI
```
a(n) = sumdiv(n, d, 6^omega(d));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 6^omega(k)*x^k/(1-x^k)))

for(n=1, 100, print1(direuler(p=2, n, (1 + 5*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

a(n) = A000005(A001014(n)).
Multiplicative with a(p^e) = 6*e+1.
a(n) = Sum_{d|n} 6^omega(d).
G.f.: Sum_{k>=1} 6^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 5/p^s). - Vaclav Kotesovec, Aug 19 2021

A344335 Number of divisors of n^8.

Original entry on oeis.org

1, 9, 9, 17, 9, 81, 9, 25, 17, 81, 9, 153, 9, 81, 81, 33, 9, 153, 9, 153, 81, 81, 9, 225, 17, 81, 25, 153, 9, 729, 9, 41, 81, 81, 81, 289, 9, 81, 81, 225, 9, 729, 9, 153, 153, 81, 9, 297, 17, 153, 81, 153, 9, 225, 81, 225, 81, 81, 9, 1377, 9, 81, 153, 49, 81, 729, 9, 153, 81, 729, 9

Offset: 1

Seiichi Manyama, May 15 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=8 of A343656.

Cf. A000005, A001016.

Table[DivisorSigma[0, n^8], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = numdiv(n^8);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 8*f[k]+1);

PARI
```
a(n) = sumdiv(n, d, 8^omega(d));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 8^omega(k)*x^k/(1-x^k)))

for(n=1, 100, print1(direuler(p=2, n, (1 + 7*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

a(n) = A000005(A001016(n)).
Multiplicative with a(p^e) = 8*e+1.
a(n) = Sum_{d|n} 8^omega(d).
G.f.: Sum_{k>=1} 8^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 7/p^s). - Vaclav Kotesovec, Aug 19 2021

A344336 Number of divisors of n^9.

Original entry on oeis.org

1, 10, 10, 19, 10, 100, 10, 28, 19, 100, 10, 190, 10, 100, 100, 37, 10, 190, 10, 190, 100, 100, 10, 280, 19, 100, 28, 190, 10, 1000, 10, 46, 100, 100, 100, 361, 10, 100, 100, 280, 10, 1000, 10, 190, 190, 100, 10, 370, 19, 190, 100, 190, 10, 280, 100, 280, 100, 100, 10, 1900, 10, 100

Offset: 1

Seiichi Manyama, May 15 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=9 of A343656.

Cf. A000005, A001017, A344337 (9^omega(n)).

Table[DivisorSigma[0, n^9], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI
```
a(n) = numdiv(n^9);
```

a(n) = prod(k=1, #f=factor(n)[, 2], 9*f[k]+1);

PARI
```
a(n) = sumdiv(n, d, 9^omega(d));
```

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, 9^omega(k)*x^k/(1-x^k)))

for(n=1, 100, print1(direuler(p=2, n, (1 + 8*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 19 2021

a(n) = A000005(A001017(n)).
Multiplicative with a(p^e) = 9*e+1.
a(n) = Sum_{d|n} 9^omega(d).
G.f.: Sum_{k>=1} 9^omega(k) * x^k/(1 - x^k).
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 8/p^s). - Vaclav Kotesovec, Aug 19 2021

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.

cp's OEIS Frontend

A343526 Number of divisors of n^7.

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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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A344327 Number of divisors of n^4.

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A344328 Number of divisors of n^5.

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A344329 Number of divisors of n^6.

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A344335 Number of divisors of n^8.

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PARI

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