A007426 -id:A007426 - cp's OEIS frontend

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

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

A360997 Multiplicative with a(p^e) = e + 3.

Original entry on oeis.org

1, 4, 4, 5, 4, 16, 4, 6, 5, 16, 4, 20, 4, 16, 16, 7, 4, 20, 4, 20, 16, 16, 4, 24, 5, 16, 6, 20, 4, 64, 4, 8, 16, 16, 16, 25, 4, 16, 16, 24, 4, 64, 4, 20, 20, 16, 4, 28, 5, 20, 16, 20, 4, 24, 16, 24, 16, 16, 4, 80, 4, 16, 20, 9, 16, 64, 4, 20, 16, 64, 4, 30, 4, 16

Offset: 1

Vaclav Kotesovec, Feb 28 2023

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

Cf. A005361 (multiplicative with a(p^e) = e), A000005 (e+1), A343443 (e+2), this sequence (e+3), A322327 (2*e), A048691 (2*e+1), A360908 (2*e-1), A226602 (3*e), A048785 (3*e+1), A360910 (3*e-1), A360909 (3*e+2), A360911 (3*e-2), A322328 (4*e), A360996 (5*e).

Cf. A000005, A007425, A007426, A074816.

g[p_, e_] := e+3; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

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

Dirichlet g.f.: Product_{primes p} (1 + (4*p^s - 3)/(p^s - 1)^2).
Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 5/p^(2*s) + 6/p^(3*s) - 2/p^(4*s)).
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A361264(n)).
a(n) = A074816(n)*A007426(n)/A007425(n). (End)

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.

A318768 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} tau(j), where tau = number of divisors (A000005).

Original entry on oeis.org

1, 2, 4, 2, 4, 8, 4, 0, 10, 8, 4, 8, 4, 8, 16, -5, 4, 20, 4, 8, 16, 8, 4, 0, 10, 8, 20, 8, 4, 32, 4, -14, 16, 8, 16, 20, 4, 8, 16, 0, 4, 32, 4, 8, 40, 8, 4, -20, 10, 20, 16, 8, 4, 40, 16, 0, 16, 8, 4, 32, 4, 8, 40, -28, 16, 32, 4, 8, 16, 32, 4, 0, 4, 8, 40, 8, 16, 32, 4

Offset: 1

Ilya Gutkovskiy, Sep 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A007425, A007426, A051062 (positions of 0's), A288571.

Cf. A001620, A082633.

Table[Sum[(-1)^(n/d + 1) Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[n]}], {n, 79}]
nmax = 79; Rest[CoefficientList[Series[Sum[DivisorSum[k, DivisorSigma[0, #] &] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 79; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSum[k, DivisorSigma[0, #] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := If[p == 2, 1 + (7 - e^2)*e/6, Binomial[e + 3, 3]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n) = sumdiv(n, d, (-1)^(n/d+1) * sumdiv(d, j, numdiv(j))); \\ Michel Marcus, Sep 04 2018

G.f.: Sum_{k>=1} tau_3(k)*x^k/(1 + x^k), where tau_3() = A007425.
L.g.f.: log(Product_{k>=1} (1 + x^k)^(tau_3(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
Multiplicative with a(2^e) = 1 + (7-e^2)*e/6, and a(p^e) = binomial(e+3,3) for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Dec 18 2023: (Start)
Dirichlet g.f.: zeta(s)^4 * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ (log(2)/2) * n * (log(n)^2 + (8 * gamma - log(2) - 2) * log(n) + 12 * gamma^2 - 8 * gamma + log(2) + 2 - 4 * gamma * log(2) + log(2)^2/3 - 8 * gamma_1), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). (End)

A341880 Number of ordered factorizations of n into 4 factors > 1.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 12, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 0, 16, 1, 0, 0, 12, 0, 0, 0, 4, 0, 12, 0, 0, 0, 0, 0, 40, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 28, 0, 0, 0, 16

Offset: 16

Ilya Gutkovskiy, Feb 22 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 16..20000
Eric Weisstein's World of Mathematics, Ordered Factorization

Column k=4 of A251683.

Cf. A000005, A007425, A007426, A070824, A074206, A200221, A341881, A341882.

b:= proc(n) option remember; series(x*(1+add(b(n/d),
      d=numtheory[divisors](n) minus {1, n})), x, 5)
    end:
a:= n-> coeff(b(n), x, 4):
seq(a(n), n=16..112);  # 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, 5}];
a[n_] := Coefficient[b[n], x, 4];
Table[a[n], {n, 16, 112}] (* Jean-François Alcover, Feb 28 2022, after Alois P. Heinz *)

Dirichlet g.f.: (zeta(s) - 1)^4.

a(n) = 6 * A000005(n) - 4 * A007425(n) + A007426(n) - 4 for n > 1.

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.

A143354 Triangle read by rows, (A051731)^4 * A128407, 1<=k<=n.

Original entry on oeis.org

1, 4, -1, 4, 0, -1, 10, -4, 0, 0, 4, 0, 0, 0, -1, 16, -4, -4, 0, 0, 1, 4, 0, 0, 0, 0, 0, -1, 20, -10, 0, 0, 0, 0, 0, 0, 10, 0, -4, 0, 0, 0, 0, 0, 0, 16, -4, 0, 0, -4, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 40, -16, -10, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 1

Gary W. Adamson, Aug 10 2008

Left border = A007426.
Right border = mu(n), A008683.
Row sums = A007425: (1, 3, 3, 6, 3, 9,...).

			First few rows of the triangle =
1;
4, -1;
4, 0, -1;
10, -4, 0, 0;
4, 0, 0, 0, 0, -1;
16, -4, -4, 0, 0, 1;
4, 0, 0, 0, 0, 0, -1;
10, -10, 0, 0, 0, 0, 0, 0;
...

Cf. A051731, A128407, A007425, A007426, A008683.

Triangle read by rows, (A051731)^4 * A128407, 1<=k<=n

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A360997 Multiplicative with a(p^e) = e + 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A127172 Cube of A051731.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A318768 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} tau(j), where tau = number of divisors (A000005).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A341880 Number of ordered factorizations of n into 4 factors > 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple