A007426 -id:A007426 - cp's OEIS frontend

A061202 (tau<=)_4(n).

1, 5, 9, 19, 23, 39, 43, 63, 73, 89, 93, 133, 137, 153, 169, 204, 208, 248, 252, 292, 308, 324, 328, 408, 418, 434, 454, 494, 498, 562, 566, 622, 638, 654, 670, 770, 774, 790, 806, 886, 890, 954, 958, 998, 1038, 1054, 1058, 1198, 1208, 1248, 1264, 1304, 1308

Offset: 1

Vladeta Jovovic, Apr 21 2001

(tau<=)_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k <= n}|, i.e., (tau<=)_k(n) is number of solutions to x_1*x_2*...*x_k <= n, x_i > 0.
Partial sums of A007426.
Equals row sums of triangle A140703. - Gary W. Adamson, May 24 2008

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Eric Weisstein's World of Mathematics, Stieltjes Constants
Wikipedia, Stieltjes constants

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_3(n): A061201, (tau<=)_5(n): A061203, (tau<=)_6(n): A061204.
Equals left column of triangle A140705.
Cf. A140703.

(* Asymptotics: *) n * (Log[n]^3/6 + (2*EulerGamma - 1/2)*Log[n]^2 + (6*EulerGamma^2 - 4*EulerGamma - 4*StieltjesGamma[1] + 1)*Log[n] + 4*EulerGamma^3 - 6*EulerGamma^2 + 4*EulerGamma + 4*StieltjesGamma[1]*(1 - 3*EulerGamma) + 2*StieltjesGamma[2] - 1) (* Vaclav Kotesovec, Sep 09 2018 *)

(tau<=)k(n) = Sum{i=1..n} tau_k(i).
a(n) = Sum_{k = 1..n} tau_{3}(k)*floor (n/k), where tau_{3} is A007425. - Enrique Pérez Herrero, Jan 23 2013
a(n) ~ n * (log(n)^3/6 + (2*g - 1/2)*log(n)^2 + (6*g^2 - 4*g - 4*g1 + 1)*log(n) + 4*g^3 - 6*g^2 + 4*g + 4*g1*(1 - 3*g) + 2*g2 - 1), where g is the Euler-Mascheroni constant A001620, g1 and g2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Sep 09 2018
a(n) = Sum_{i=1..n} tau(i)*A006218(floor(n/i)). - Ridouane Oudra, Sep 17 2021
a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} floor(n/(i*j*k)). - Ridouane Oudra, Oct 31 2022

A280486 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijk*l)).

Original entry on oeis.org

1, 1, 4, 8, 20, 36, 86, 150, 314, 564, 1088, 1902, 3557, 6085, 10902, 18506, 32124, 53584, 91133, 149749, 249315, 405121, 662582, 1063152, 1714580, 2719842, 4327302, 6797316, 10686005, 16622003, 25861855, 39866017, 61422891, 93910783, 143406552, 217537696

Offset: 0

Vaclav Kotesovec, Jan 04 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000009, A007426, A107742, A280473, A280487, A219561.

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j*k*l)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}, {l, 1, nmax/i/j/k}], {x, 0, nmax}], x]
nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#] * DivisorSigma[0, #] &], {n, 1, nmax}]; s = 1 + x; Do[s *= Sum[Binomial[tau4[[k]], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Sep 08 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^tau_4(k), where tau_4() = A007426. - Ilya Gutkovskiy, May 22 2018

A061203 (tau<=)_5(n).

Original entry on oeis.org

1, 6, 11, 26, 31, 56, 61, 96, 111, 136, 141, 216, 221, 246, 271, 341, 346, 421, 426, 501, 526, 551, 556, 731, 746, 771, 806, 881, 886, 1011, 1016, 1142, 1167, 1192, 1217, 1442, 1447, 1472, 1497, 1672, 1677, 1802, 1807, 1882, 1957, 1982, 1987, 2337, 2352

Offset: 1

Vladeta Jovovic, Apr 21 2001

(tau<=)_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k <= n}|, i.e., (tau<=)_k(n) is number of solutions to x_1*x_2*...*x_k <= n, x_i > 0.
Partial sums of A061200.
Equals row sums of triangle A140705. - Gary W. Adamson, May 24 2008

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_3(n): A061201, (tau<=)_4(n): A061202, (tau<=)_6(n): A061204.

Cf. A140705.

b:= proc(k, n) option remember; uses numtheory;
     `if`(k=1, 1, add(b(k-1, d), d=divisors(n)))
    end:
a:= proc(n) option remember; `if`(n=0, 0, b(5, n)+a(n-1)) end:
seq(a(n), n=1..49);  # Alois P. Heinz, Feb 13 2022

nmax = 50;
tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #] &], {n, 1, nmax}];
Accumulate[Table[Sum[tau4[[d]], {d, Divisors[n]}], {n, nmax}]] (* Vaclav Kotesovec, Sep 10 2018 *)

(tau<=)k(n) = Sum{i=1..n} tau_k(i).
a(n) = Sum_{k=1..n} tau_{4}(k) * floor(n/k), where tau_{4} is A007426. - Enrique Pérez Herrero, Jan 23 2013
a(n) ~ n*(log(n)^4/24 + (5*g/6 - 1/6)*log(n)^3 + 10*g1^2 + (5*g^2 - 5*g/2 - 5*g1/2 + 1/2)*log(n)^2 + (10*g^3 - 10*g^2 + (5 - 20*g1)*g + 5*g1 + 5*g2/2 - 1)*log(n) + 5*g^4 - 10*g^3 + (10 - 30*g1)*g^2 + (20*g1 + 10*g2 - 5)*g - 5*g1 - 5*g2/2 - 5*g3/6 + 1), where g is the Euler-Mascheroni constant A001620 and g1, g2, g3 are the Stieltjes constants, see A082633, A086279 and A086280. - Vaclav Kotesovec, Sep 10 2018

A061204 (tau<=)_6(n).

Original entry on oeis.org

1, 7, 13, 34, 40, 76, 82, 138, 159, 195, 201, 327, 333, 369, 405, 531, 537, 663, 669, 795, 831, 867, 873, 1209, 1230, 1266, 1322, 1448, 1454, 1670, 1676, 1928, 1964, 2000, 2036, 2477, 2483, 2519, 2555, 2891, 2897, 3113, 3119, 3245, 3371, 3407, 3413

Offset: 1

Vladeta Jovovic, Apr 21 2001

(tau<=)_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k<=n}|, i.e. (tau<=)_k(n) is number of solutions to x_1*x_2*...*x_k<=n, x_i>0.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_3(n): A061201, (tau<=)_4(n): A061202, (tau<=)_5(n): A061203.

nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #] &], {n, 1, nmax}]; tau5 = Table[Sum[tau4[[d]], {d, Divisors[n]}], {n, nmax}]; Accumulate[Table[Sum[tau5[[d]], {d, Divisors[n]}], {n, nmax}]] (* Vaclav Kotesovec, Sep 10 2018 *)

(tau<=)k(n)=Sum{i=1..n} tau_k(i). a(n)=partial sums of A034695.
a(n) = Sum_{k=1..n} tau_{5}(k) * floor(n/k), where tau_{5} is A061200. - Enrique Pérez Herrero, Jan 23 2013
a(n) ~ n*(log(n)^5/120 + (g/4 - 1/24)*log(n)^4 + (5*g^2/2 - g - g1 + 1/6)*log(n)^3 + (10*g^3 - 15*g^2/2 + (3 - 15*g1)*g + 3*g1 + 3*g2/2 - 1/2)*log(n)^2 + (15*g^4 - 20*g^3 + (15 - 60*g1)*g^2 + (30*g1 + 15*g2 - 6)*g + 15*g1^2 - 6*g1 - 3*g2 - g3 + 1)*log(n) + 6*g^5 - 15*g^4 + (20 - 60*g1)*g^3 + (60*g1 + 30*g2 - 15)*g^2 + (60*g1^2 - 30*g1 - 15*g2 - 5*g3 + 6)*g - 15*g1^2 + g1*(6 - 15*g2) + 3*g2 + g3 + g4/4 - 1), where g is the Euler-Mascheroni constant A001620 and g1, g2, g3, g4 are the Stieltjes constants, see A082633, A086279, A086280 and A086281. - Vaclav Kotesovec, Sep 10 2018

A111217 d_7(n), tau_7(n), number of ordered factorizations of n as n = rstuvwx (7-factorizations).

Original entry on oeis.org

1, 7, 7, 28, 7, 49, 7, 84, 28, 49, 7, 196, 7, 49, 49, 210, 7, 196, 7, 196, 49, 49, 7, 588, 28, 49, 84, 196, 7, 343, 7, 462, 49, 49, 49, 784, 7, 49, 49, 588, 7, 343, 7, 196, 196, 49, 7, 1470, 28, 196, 49, 196, 7, 588, 49, 588, 49, 49, 7, 1372, 7, 49, 196, 924, 49, 343, 7, 196

Offset: 1

Gerald McGarvey, Oct 25 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)

Cf. tau_k(n) for k>=2: A000005, A007425, A007426, A061200, A034695, A111218 - A111221, A111306.

Column k=7 of A077592.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 7], {n, 68}] (* Robert G. Wilson v, Nov 02 2005 *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 7], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)

for(n=1,100,print1(sumdiv(n,i,sumdiv(i,j,sumdiv(j,k,sumdiv(k,l,sumdiv(l,x,numdiv(x)))))),","))

a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+6, 6)) \\ Charles R Greathouse IV, Oct 28 2017

Dirichlet convolution of A000012 with A034695, or of A000005 with A061200, or of A007425 with A007426. Dirichlet g.f. zeta^7(s). - R. J. Mathar, Apr 01 2011
G.f.: Sum_{k>=1} tau_6(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018
Multiplicative with a(p^e) = binomial(e+6,6). - Amiram Eldar, Sep 13 2020

A111219 d_9(n), tau_9(n), number of ordered factorizations of n as n = rstuvwxyz (9-factorizations).

Original entry on oeis.org

1, 9, 9, 45, 9, 81, 9, 165, 45, 81, 9, 405, 9, 81, 81, 495, 9, 405, 9, 405, 81, 81, 9, 1485, 45, 81, 165, 405, 9, 729, 9, 1287, 81, 81, 81, 2025, 9, 81, 81, 1485, 9, 729, 9, 405, 405, 81, 9, 4455, 45, 405, 81, 405, 9, 1485, 81, 1485, 81, 81, 9, 3645, 9, 81, 405, 3003, 81

Offset: 1

Gerald McGarvey, Oct 25 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)

Cf. tau_2(n)...tau_6(n): A000005, A007425, A007426, A061200, A034695.

Column k=9 of A077592.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 9], {n, 65}] (* Robert G. Wilson v, Nov 02 2005 *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 9], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)

for(n=1,100,print1(sumdiv(n,i,sumdiv(i,j,sumdiv(j,k,sumdiv(k,l,sumdiv(l,m,sumdiv(m,o,sumdiv(o,x,numdiv(x)))))))),","))

a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+8, 8)) \\ Charles R Greathouse IV, Oct 28 2017

G.f.: Sum_{k>=1} tau_8(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+8,8). - Amiram Eldar, Sep 13 2020

A111220 d_10(n), tau_10(n), number of ordered factorizations of n as n = rstuvwxyza (10-factorizations).

Original entry on oeis.org

1, 10, 10, 55, 10, 100, 10, 220, 55, 100, 10, 550, 10, 100, 100, 715, 10, 550, 10, 550, 100, 100, 10, 2200, 55, 100, 220, 550, 10, 1000, 10, 2002, 100, 100, 100, 3025, 10, 100, 100, 2200, 10, 1000, 10, 550, 550, 100, 10, 7150, 55, 550, 100, 550, 10, 2200, 100

Offset: 1

Gerald McGarvey, Oct 25 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.

Cf. tau_2(n)...tau_6(n): A000005, A007425, A007426, A061200, A034695.

Column k=10 of A077592.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 10], {n, 55}] (* Robert G. Wilson v, Nov 02 2005 *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 10], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)

for(n=1,100,print1(sumdiv(n,i,sumdiv(i,j,sumdiv(j,k,sumdiv(k,l,sumdiv(l,m,sumdiv(m,o,sumdiv(o,p,sumdiv(p,x,numdiv(x))))))))),","))

a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+9, 9)) \\ Charles R Greathouse IV, Oct 28 2017

G.f.: Sum_{k>=1} tau_9(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+9,9). - Amiram Eldar, Sep 13 2020

A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(ijk*l)).

Original entry on oeis.org

1, 1, 5, 9, 29, 49, 135, 235, 565, 995, 2177, 3821, 7900, 13728, 26974, 46606, 88128, 150644, 276283, 467647, 835708, 1400874, 2448818, 4065230, 6975307, 11470265, 19359345, 31552473, 52488142, 84808548, 139274675, 223191639, 362297234, 576064732, 925295844

Offset: 0

Vaclav Kotesovec, Jan 04 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000041, A006171, A007426, A174465, A280486.

nmax=50; CoefficientList[Series[Product[1/(1-x^(i*j*k*l)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}, {l, 1, nmax/i/j/k}], {x, 0, nmax}], x]
nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#] * DivisorSigma[0, #] &], {n, 1, nmax}]; s = 1 - x; Do[s *= Sum[Binomial[tau4[[k]], j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2018 *)

G.f.: Product_{k>=1} 1/(1 - x^k)^tau_4(k), where tau_4() = A007426. - Ilya Gutkovskiy, May 22 2018

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}]

A062367 Multiplicative with a(p^e) = (e+1)(e+2)(2*e+3)/6.

Original entry on oeis.org

1, 5, 5, 14, 5, 25, 5, 30, 14, 25, 5, 70, 5, 25, 25, 55, 5, 70, 5, 70, 25, 25, 5, 150, 14, 25, 30, 70, 5, 125, 5, 91, 25, 25, 25, 196, 5, 25, 25, 150, 5, 125, 5, 70, 70, 25, 5, 275, 14, 70, 25, 70, 5, 150, 25, 150, 25, 25, 5, 350, 5, 25, 70, 140, 25, 125, 5, 70, 25, 125, 5

Offset: 1

Vladeta Jovovic, Jul 07 2001

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A000012, A000330, A029939, A035116, A048691, A060648, A062368.

A062367 := proc(n)
    add(numtheory[tau](d)^2,d=numtheory[divisors](n)) ;
end proc:
seq(A062367(n),n=1..40) ; # R. J. Mathar, May 15 2025

{1}~Join~Array[Times @@ Map[((# + 1) (# + 2) (2 # + 3))/6 &, FactorInteger[#][[All, -1]] ] &, 70, 2] (* or *)
Array[DivisorSum[#, DivisorSigma[0, #]^2 &] &, 71] (* Michael De Vlieger, Mar 05 2021 *)

a(n) = sumdiv(n, d, numdiv(d)^2) \\ Michel Marcus, Jun 17 2013

a(n) = Sum_{i|n, j|n} tau(gcd(i, j)) = Sum_{d|n} tau(d)^2.
a(n) = Sum_{i|n, j|n} tau(i)*tau(j)/tau(lcm(i, j)), where tau(n) = number of divisors of n, cf. A000005.
Dirichlet convolution of A035116 and A000012 (i.e., inverse Mobius transform of A035116). Dirichlet g.f.: zeta^5(s)/zeta(2s). - R. J. Mathar, Feb 03 2011
G.f.: Sum_{n>=1} A000005(n)^2*x^n/(1-x^n). - Mircea Merca, Feb 26 2014
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(tau(k)^2/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
Dirichlet convolution of A007426 and A008966. Dirichlet convolution of A007425 and A034444. - R. J. Mathar, Jun 05 2020
Let b(n), n > 0, be Dirichlet inverse of a(n). Then b(n) is multiplicative with b(p^e) = (-1)^e*(Sum_{i=0..e} binomial(4,i)) for prime p and e >= 0, where binomial(n,k)=0 if n < k; abs(b(n)) is multiplicative and has the Dirichlet g.f.: (zeta(s))^5/(zeta(2*s))^4. - Werner Schulte, Feb 07 2021
a(n) = Sum_{d divides n} tau(d^2)*tau(n/d), Dirichlet convolution of A048691 and A000005. - Peter Bala, Jan 26 2024

cp's OEIS Frontend

A061202 (tau<=)_4(n).

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A280486 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(i*j*k*l)).

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Mathematica

Formula

A061203 (tau<=)_5(n).

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Maple

Mathematica

Formula

A061204 (tau<=)_6(n).

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Mathematica

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A111217 d_7(n), tau_7(n), number of ordered factorizations of n as n = rstuvwx (7-factorizations).

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Mathematica

PARI

PARI

Formula

A111219 d_9(n), tau_9(n), number of ordered factorizations of n as n = rstuvwxyz (9-factorizations).

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Mathematica

PARI

PARI

Formula

A111220 d_10(n), tau_10(n), number of ordered factorizations of n as n = rstuvwxyza (10-factorizations).

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Mathematica

PARI

PARI

Formula

A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(i*j*k*l)).

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A280486 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijk*l)).

A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(ijk*l)).

A062367 Multiplicative with a(p^e) = (e+1)(e+2)(2*e+3)/6.