A061200 -id:A061200 - cp's OEIS frontend

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

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

A082476 a(n) = Sum_{d|n} mu(d)^2*tau(d)^2.

Original entry on oeis.org

1, 5, 5, 5, 5, 25, 5, 5, 5, 25, 5, 25, 5, 25, 25, 5, 5, 25, 5, 25, 25, 25, 5, 25, 5, 25, 5, 25, 5, 125, 5, 5, 25, 25, 25, 25, 5, 25, 25, 25, 5, 125, 5, 25, 25, 25, 5, 25, 5, 25, 25, 25, 5, 25, 25, 25, 25, 25, 5, 125, 5, 25, 25, 5, 25, 125, 5, 25, 25, 125, 5, 25, 5, 25, 25, 25, 25, 125

Offset: 1

Benoit Cloitre, Apr 27 2003

More generally : sum(d|n, mu(d)^2*tau(d)^m) = (2^m+1)^omega(n).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000351, A001221, A007425, A007947, A008683, A061200, A074816.

tau[1, n_] := 1; SetAttributes[tau, Listable];
tau[k_, n_] := Plus @@ (tau[k - 1, Divisors[n]]) /; k > 1;
A082476[n_] := Abs[DivisorSum[n, MoebiusMu[ # ]*tau[3, #^2] &]]; (* Enrique Pérez Herrero, Mar 29 2010 *)
(* or more easy *)
A082476[n_] := 5^PrimeNu[n] (* Enrique Pérez Herrero, Mar 29 2010 *)

PARI
```
a(n)=5^omega(n)
```

for(n=1, 100, print1(direuler(p=2, n, (4*X+1)/(1-X))[n], ", ")) \\ Vaclav Kotesovec, Feb 28 2023

a(n) = 5^omega(n); multiplicative with a(p^e)=5.
a(n) = abs(sum(d|n, mu(d)*tau_3(d^2))), where tau_3 is A007425. - Enrique Pérez Herrero, Mar 29 2010
a(n) = tau_5(rad(n)) = A061200(A007947(n)). - Enrique Pérez Herrero, Jun 24 2010
a(n) = A000351(A001221(n)). - Antti Karttunen, Jul 26 2017
From Vaclav Kotesovec, Feb 28 2023: (Start)
Dirichlet g.f.: Product_{primes p} (1 + 5/(p^s - 1)).
Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 10/p^(2*s) + 20/p^(3*s) - 15/p^(4*s) + 4/p^(5*s)), (with a product that converges for s=1). (End)

A111221 d_11(n), tau_11(n), number of ordered factorizations of n as n = rstuvwxyzab (11-factorizations).

Original entry on oeis.org

1, 11, 11, 66, 11, 121, 11, 286, 66, 121, 11, 726, 11, 121, 121, 1001, 11, 726, 11, 726, 121, 121, 11, 3146, 66, 121, 286, 726, 11, 1331, 11, 3003, 121, 121, 121, 4356, 11, 121, 121, 3146, 11, 1331, 11, 726, 726, 121, 11, 11011, 66, 726, 121, 726, 11, 3146, 121

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_1(n): A000012
Cf. tau_2(n)...tau_6(n): A000005, A007425, A007426, A061200, A034695.
Cf. tau_7(n)...tau_10(n): A111217, A111218, A111219, A111220.
Cf. tau_12(n): A111306.
Column k=11 of A077592.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 11], {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, 11], {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,q,sumdiv(q,x,numdiv(x)))))))))),","))

a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+10, 10)) \\ Charles R Greathouse IV, Oct 28 2017

G.f.: Sum_{k>=1} tau_10(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+10,10). - Amiram Eldar, Sep 13 2020

A111306 d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).

Original entry on oeis.org

1, 12, 12, 78, 12, 144, 12, 364, 78, 144, 12, 936, 12, 144, 144, 1365, 12, 936, 12, 936, 144, 144, 12, 4368, 78, 144, 364, 936, 12, 1728, 12, 4368, 144, 144, 144, 6084, 12, 144, 144, 4368, 12, 1728, 12, 936, 936, 144, 12, 16380, 78, 936, 144, 936, 12, 4368, 144

Offset: 1

Gerald McGarvey, Nov 02 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_11(n): A000005, A007425, A007426, A061200, A034695, A111217, A111218, A111219, A111220, A111221.

Column k=12 of A077592.

b:= proc(n, k) option remember; `if`(k=1, 1,
      add(b(d, k-1), d=numtheory[divisors](n)))
    end:
a:= n-> b(n, 12):
seq(a(n), n=1..55);  # Alois P. Heinz, Jun 12 2024

tau[k_,1]:=1; tau[k_,n_]:=Times@@(Binomial[#+k-1,k-1]&/@FactorInteger[n][[All,2]]); Table[tau[12,n],{n,1000}] (* Enrique Pérez Herrero, Jan 17 2013 *)

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,q,sumdiv(q,r,sumdiv(r,x,numdiv(x))))))))))),","))

a(n,f=factor(n))=f=f[,2]; prod(i=1,#f, binomial(f[i]+11, 11)) \\ Charles R Greathouse IV, Oct 28 2017

G.f.: Sum_{k>=1} tau_11(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

Multiplicative with a(p^e) = binomial(e+11,11). - Amiram Eldar, Sep 13 2020

A127172 Cube of A051731.

Original entry on oeis.org

1, 3, 1, 3, 0, 1, 6, 3, 0, 1, 3, 0, 0, 0, 1, 9, 3, 3, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 10, 6, 0, 3, 0, 0, 0, 1, 6, 0, 3, 0, 0, 0, 0, 0, 1, 9, 3, 0, 0, 0, 3, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Nonzero terms in every column = A007425: (1, 3, 3, 6, 3, 9, 3, ...).
Row sums = A007426: (1, 4, 4, 20, 4, 16, ...).
A127172 * mu(n) = d(n); or A127172 * A008683 = A000005.
A127172 * d(n) = tau_5(n); or A127172 * A000005 = A061200.
A127172 * phi(n) = A007429: (1, 4, 5, 11, 7, 20, ...); or: A127172 * A000010 = A007429.
Note that A051731 * d(n) = row sums of A127172; or A051731 * A000005 = A007425.
Also, A126988 * mu(n) = phi(n); or A126988 * A008683 = A000010.
A126988 * phi(n) = A018804: (1, 3, 5, 8, 9, 15, ...); = A127170 * mu(n).

			First few rows of the triangle:
   1;
   3, 1;
   3, 0, 1;
   6, 3, 0, 1;
   3, 0, 0, 0, 1;
   9, 3, 3, 0, 0, 1;
   3, 0, 0, 0, 0, 0, 1;
  10, 6, 0, 3, 0, 0, 0, 1;
   6, 0, 3, 0, 0, 0, 0, 0, 1;
   9, 3, 0, 0, 3, 0, 0, 0, 0, 1;
  ...

Cf. A000005, A007425, A127170, A051731, A007429, A000010, A126988, A018804.

Cube of A051731 A007425: (1, 3, 3, 6, 3, 9, 3, ...) in every column k, interspersed with (k-1) zeros.

A341881 Number of ordered factorizations of n into 5 factors > 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 15

Offset: 32

Ilya Gutkovskiy, Feb 22 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 32..20000
Eric Weisstein's World of Mathematics, Ordered Factorization

Column k=5 of A251683.

Cf. A000005, A007425, A007426, A061200, A070824, A074206, A200221, A341880, A341882.

b:= proc(n) option remember; series(x*(1+add(b(n/d),
      d=numtheory[divisors](n) minus {1, n})), x, 6)
    end:
a:= n-> coeff(b(n), x, 5):
seq(a(n), n=32..128);  # Alois P. Heinz, Feb 22 2021

b[n_] := b[n] = Series[x*(1 + Sum[b[n/d],
     {d, Divisors[n] ~Complement~ {1, n}}]), {x, 0, 6}];
a[n_] := Coefficient[b[n], x, 5];
Table[a[n], {n, 32, 128}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)

Dirichlet g.f.: (zeta(s) - 1)^5.

a(n) = -10 * A000005(n) + 10 * A007425(n) - 5 * A007426(n) + A061200(n) + 5 for n > 1.

A341882 Number of ordered factorizations of n into 6 factors > 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6

Offset: 64

Ilya Gutkovskiy, Feb 22 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 64..20000
Eric Weisstein's World of Mathematics, Ordered Factorization

Column k=6 of A251683.

Cf. A000005, A007425, A007426, A034695, A061200, A070824, A074206, A200221, A341880, A341881.

b:= proc(n) option remember; series(x*(1+add(b(n/d),
      d=numtheory[divisors](n) minus {1, n})), x, 7)
    end:
a:= n-> coeff(b(n), x, 6):
seq(a(n), n=64..160);  # Alois P. Heinz, Feb 22 2021

b[n_] := b[n] = Series[x*(1 + Sum[b[n/d],
     {d, Divisors[n]~Complement~{1, n}}]), {x, 0, 7}];
a[n_] := Coefficient[b[n], x, 6];
Table[a[n], {n, 64, 160}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)

Dirichlet g.f.: (zeta(s) - 1)^6.

a(n) = 15 * A000005(n) - 20 * A007425(n) + 15 * A007426(n) - 6 * A061200(n) + A034695(n) - 6 for n > 1.

A383657 Numerator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(3/2).

Original entry on oeis.org

1, 3, 3, 15, 3, 9, 3, 35, 15, 9, 3, 45, 3, 9, 9, 315, 3, 45, 3, 45, 9, 9, 3, 105, 15, 9, 35, 45, 3, 27, 3, 693, 9, 9, 9, 225, 3, 9, 9, 105, 3, 27, 3, 45, 45, 9, 3, 945, 15, 45, 9, 45, 3, 105, 9, 105, 9, 9, 3, 135, 3, 9, 45, 3003, 9, 27, 3, 45, 9, 27, 3, 525, 3

Offset: 1

Vaclav Kotesovec, May 04 2025

In general, for m > 0, if Dirichlet g.f. is zeta(s)^m, then Sum_{j=1..n} a(j) ~ n*log(n)^(m-1)/Gamma(m) * (1 + (m-1)*(m*gamma - 1)/log(n)), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A383658.
Cf. A046643, A046644, A256688, A256689, A256690, A256691, A256692, A256693.
Cf. A000005, A007425, A007426, A061200, A034695.

coeff=CoefficientList[Series[1/(1-x)^(3/2),{x,0,20}]//Normal,x];dptTerm[n_]:=Module[{flist=FactorInteger[n]},If[n==1,coeff[[1]],Numerator[Times@@(coeff[[flist[[All,2]]+1]])]]];Array[dptTerm,73] (* Shenghui Yang, May 04 2025 *)

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(3/2))[n]), ", "))

Sum_{k=1..n} A383657(k)/A383658(k) ~ 2*n*sqrt(log(n)/Pi) * (1 - (1 - 3*gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620.

A383658 Denominator of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)^(3/2).

Original entry on oeis.org

1, 2, 2, 8, 2, 4, 2, 16, 8, 4, 2, 16, 2, 4, 4, 128, 2, 16, 2, 16, 4, 4, 2, 32, 8, 4, 16, 16, 2, 8, 2, 256, 4, 4, 4, 64, 2, 4, 4, 32, 2, 8, 2, 16, 16, 4, 2, 256, 8, 16, 4, 16, 2, 32, 4, 32, 4, 4, 2, 32, 2, 4, 16, 1024, 4, 8, 2, 16, 4, 8, 2, 128, 2, 4, 16, 16, 4

Offset: 1

Vaclav Kotesovec, May 04 2025

Is this a duplicate of A046644 (the first 8192 entries are the same)? - R. J. Mathar, May 06 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A383657.
Cf. A046643, A046644, A256688, A256689, A256690, A256691, A256692, A256693.
Cf. A000005, A007425, A007426, A061200, A034695.

coeff=CoefficientList[Series[1/(1-x)^(3/2),{x,0,20}]//Normal,x]; dptTerm[n_]:=Module[{flist=FactorInteger[n]},If[n==1,coeff[[1]],Denominator[Times@@(coeff[[flist[[All,2]]+1]])]]];Array[dptTerm,77] (* Shenghui Yang, May 04 2025 *)

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-X)^(3/2))[n]), ", "))

Sum_{k=1..n} A383657(k)/A383658(k) ~ 2*n*sqrt(log(n)/Pi) * (1 - (1 - 3*gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

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

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

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A082476 a(n) = Sum_{d|n} mu(d)^2*tau(d)^2.

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A111221 d_11(n), tau_11(n), number of ordered factorizations of n as n = rstuvwxyzab (11-factorizations).

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A111306 d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).

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A127172 Cube of A051731.

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A341881 Number of ordered factorizations of n into 5 factors > 1.

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A341882 Number of ordered factorizations of n into 6 factors > 1.

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