A073017 -id:A073017 - cp's OEIS frontend

A045778 Number of factorizations of n into distinct factors greater than 1.

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 5, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2, 5, 1, 9, 2, 3, 2, 2, 2, 10, 1, 3, 3, 5, 1, 5, 1, 5

Offset: 1

David W. Wilson

This sequence depends only on the prime signature of n and not on the actual value of n.
Also the number of strict multiset partitions (sets of multisets) of the prime factors of n. - Gus Wiseman, Dec 03 2016
Number of sets of integers greater than 1 whose product is n. - Antti Karttunen, Feb 20 2024

			24 can be factored as 24, 2*12, 3*8, 4*6, or 2*3*4, so a(24) = 5. The factorization 2*2*6 is not permitted because the factor 2 is present twice. a(1) = 1 represents the empty factorization.

David W. Wilson, Table of n, a(n) for n = 1..10000
Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, Preprint, 2012.
P. A. J. G. Chevalier, On a Mathematical Method for Discovering Relations Between Physical Quantities: a Photonics Case Study, Slides from a talk presented at ICOL2014.
P. A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium.
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1), 2006.
R. J. Mathar, Factorizations of n = 1..1500
Eric Weisstein's World of Mathematics, Unordered Factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A045779, A045780, A050323. a(p^k) = A000009. a(A002110) = A000110.
Cf. A036469, A114591, A114592, A316441 (Dirichlet inverse).
Cf. A156648 (2*Dgf at s=2), A073017 (2*Dgf at s=3), A258870 (2*Dgf at s=4).
Cf. also A069626 (Number of sets of integers > 1 whose least common multiple is n).
Cf. A287549 (partial sums).

⍝ Dyalog dialect
divisors ← {ð←⍵{(0=⍵|⍺)/⍵}⍳⌊⍵*÷2 ⋄ 1=⍵:ð ⋄ ð, (⍵∘÷)¨(⍵=(⌊⍵*÷2)*2)↓⌽ð}
A045778 ← { D←1↓divisors(⍵) ⋄ T←(⍴D)⍴2 ⋄ +/⍵⍷{×/D/⍨T⊤⍵}¨(-∘1)⍳2*⍴D } ⍝ (simple, but a memory hog)
A045778 ← { ⍺←⌽divisors(⍵) ⋄ 1=⍵:1 ⋄ 0=≢⍺:0 ⋄ R←⍺↓⍨⍺⍳⍵∘÷ ⋄ Ð←{⍺/⍨0=⍺|⍵} ⋄ +/(((R)Ð⊢)∇⊢)¨(⍵∘÷)¨⍺ } ⍝ (more efficient) - Antti Karttunen, Feb 20 2024

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..120);  # Alois P. Heinz, May 26 2013

gd[m_, 1] := 1; gd[1, n_] := 0; gd[1, 1] := 1; gd[0, n_] := 0; gd[m_, n_] := gd[m, n] = Total[gd[# - 1, n/#] & /@ Select[Divisors[n], # <= m &]]; Array[ gd[#, #] &, 100]  (* Alexander Adam, Dec 28 2012 *)

v=vector(100,k,k==1); for(n=2,#v, v+=dirmul(v,vector(#v,k,k==n)) ); v /* Max Alekseyev, Jul 16 2014 */

A045778(n, k=n) = ((n<=k) + sumdiv(n, d, if(d > 1 && d <= k && d < n, A045778(n/d, d-1)))); \\ After Alois P. Heinz's Maple-code by Antti Karttunen, Jul 23 2017, edited Feb 20 2024

A045778(n, m=n) = if(1==n, 1, sumdiv(n,d,if((d>1)&&(d<=m),A045778(n/d,d-1)))); \\ Antti Karttunen, Feb 20 2024

from sympy.core.cache import cacheit
from sympy import divisors, isprime
@cacheit
def b(n, k): return (0 if n>k else 1) + (0 if isprime(n) else sum(0 if d>k else b(n//d, d - 1) for d in divisors(n)[1:-1]))
def a(n): return b(n, n)
print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Aug 19 2017, after Maple code

Dirichlet g.f.: Product_{n>=2} (1 + 1/n^s).
Let p and q be two distinct prime numbers and k a natural number. Then a(p^k) = A000009(k) and a(p^k*q) = A036469(k). - Alexander Adam, Dec 28 2012
Let p_i with 1<=i<=k k distinct prime numbers. Then a(Product_{i=1..k} p_i) = A000110(k). - Alexander Adam, Dec 28 2012

Edited by Franklin T. Adams-Watters, Jun 04 2009

A156648 Decimal expansion of Product_{k>=1} (1 + 1/k^2).

Original entry on oeis.org

3, 6, 7, 6, 0, 7, 7, 9, 1, 0, 3, 7, 4, 9, 7, 7, 7, 2, 0, 6, 9, 5, 6, 9, 7, 4, 9, 2, 0, 2, 8, 2, 6, 0, 6, 6, 6, 5, 0, 7, 1, 5, 6, 3, 4, 6, 8, 2, 7, 6, 3, 0, 2, 7, 7, 4, 7, 8, 0, 0, 3, 5, 9, 3, 5, 5, 7, 4, 4, 7, 3, 2, 4, 1, 1, 1, 0, 2, 2, 0, 7, 3, 2, 1, 3, 2, 5, 5, 9, 2, 6, 5, 9, 0, 3, 2, 3, 0, 2, 3, 5, 2, 8, 7, 5

Offset: 1

R. J. Mathar, Feb 12 2009

Consider the value at s = 2 of the partition zeta functions zeta_{type}(s), where the defining sum runs over partitions into 'type' parts, where 'type' is 'even', 'prime' or 'distinct'. (For the precise definitions see R. Schneider's dissertation.) Then
  zeta_{even}(2) = Pi/2 = A019669;
  zeta_{prime}(2) = Pi^2/6 = A013661;
  zeta_{distinct}(2) = sinh(Pi)/Pi, this constant. - Peter Luschny, Aug 11 2021
For m>0, Product_{k>=1} (1 + m/k^2) = sinh(Pi*sqrt(m)) / (Pi*sqrt(m)). - Vaclav Kotesovec, Aug 30 2024

			3.676077910374977720695697492028260666507156346827630277478003593557447324111... = (1+1)*(1+1/4)*(1+1/9)*(1+1/16)*(1+1/25)*...

Reinhold Remmert, Classical topics in complex function theory, Vol. 172 of Graduate Texts in Mathematics, p. 12, Springer, 1997.

Robert Schneider, Eulerian series, zeta functions and the arithmetic of partitions, arXiv:2008.04243 [math.NT], 2020.

Square root of A084243.

Cf. A084243, A073017, A258870, A258871, A334411, A019669, A013661.

Maple
```
evalf(sinh(Pi)/Pi) ;
```

RealDigits[Sinh[Pi]/Pi, 10, 111][[1]] (* or *)
RealDigits[Re[1/(I!*(-I)!)], 10, 111][[1]] (* Robert G. Wilson v, Feb 23 2015 *)

sinh(Pi)/Pi \\ Charles R Greathouse IV, Dec 16 2013

prodnumrat(1 + 1/x^2, 1) \\ Charles R Greathouse IV, Feb 03 2025

Equals sinh(Pi)/Pi.
Equals 1/A090986. - R. J. Mathar, Mar 05 2009
Binomial(2, 1+i) = 1/(i!*(-i)!) (where x! means Gamma(x+1)). - Robert G. Wilson v, Feb 23 2015
Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(2*j)/j)). - Vaclav Kotesovec, Mar 28 2019
Equals Product_{k>=1} (1+2/(k*(k+2))). - Amiram Eldar, Aug 16 2020

A109219 Decimal expansion of Product_{n >= 2} 1-n^(-3).

Original entry on oeis.org

8, 0, 9, 3, 9, 6, 5, 9, 7, 3, 6, 6, 2, 9, 0, 1, 0, 9, 5, 7, 8, 6, 8, 0, 4, 7, 8, 7, 2, 6, 3, 8, 2, 1, 1, 9, 3, 7, 2, 7, 8, 7, 6, 4, 8, 2, 6, 1, 1, 3, 0, 1, 6, 5, 8, 7, 7, 5, 8, 3, 3, 2, 4, 9, 0, 8, 8, 1, 4, 9, 1, 1, 3, 7, 3, 6, 2, 2, 7, 8, 9, 3, 7, 4, 6, 0, 1, 8, 3, 3, 8, 5, 7, 3, 5, 3, 0, 1, 4, 6, 2, 7, 1, 2, 6

Offset: 0

Zak Seidov, Apr 17 2006

The physical applications of this kind of product (with s<0) can be found in the Klauder et al. reference. - Karol A. Penson, Feb 24 2006

			0.809396597366290109578680478726382119372787648261130...

J. R. Klauder, K. A. Penson and J.-M. Sixdeniers, Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems, Physical Review A, Vol. 64, p. 013817 (2001).

Cf. A073017, A175615, A175617, A175619, A258870.

RealDigits[N[Cosh[(Sqrt[3]*Pi)/2]/(3*Pi), 150]] (* T. D. Noe, Apr 24 2006 *)

exp(suminf(j=1, (1 - zeta(3*j))/j)) \\ Vaclav Kotesovec, Apr 27 2020

prodnumrat(1-1/x^3,2) \\ Charles R Greathouse IV, Feb 03 2025

Equals cosh((sqrt(3)*Pi)/2)/(3*Pi).
Product_{n >= 2} (1 - 1/n^p) simplifies, if p is odd, to 1/(p * Product_{j=1..p-1} Gamma(-(-1)^(j*(1 + 1/p)))) and, if p is even, to the elementary (Product_{j=1..p/2-1} sin(Pi*(-1)^(2*j/p))/(Pi*i)) / p. - David W. Cantrell, Feb 24 2006
Equals exp(Sum_{j>=1} (1 - zeta(3*j))/j). - Vaclav Kotesovec, Apr 27 2020
Equals 1/(Gamma((5-i*sqrt(3))/2)*Gamma((5+i*sqrt(3))/2)). - Amiram Eldar, Sep 01 2020

Corrected and extended by T. D. Noe, Apr 24 2006

A258870 Decimal expansion of Product_{n>=1} (1+1/n^4).

Original entry on oeis.org

2, 1, 6, 7, 3, 6, 0, 6, 2, 5, 8, 8, 2, 2, 6, 1, 9, 5, 1, 9, 0, 0, 2, 3, 1, 3, 6, 6, 8, 4, 7, 0, 2, 7, 4, 4, 1, 8, 2, 1, 6, 1, 3, 1, 7, 2, 9, 6, 3, 4, 9, 8, 5, 0, 9, 7, 5, 6, 2, 3, 2, 5, 9, 9, 8, 8, 2, 2, 1, 3, 7, 8, 7, 1, 9, 4, 8, 5, 3, 8, 1, 6, 7, 7, 0, 4, 2, 6, 8, 1, 2, 3, 6, 4, 1, 5, 4, 4, 4, 7, 3, 7, 9, 5, 0, 3, 6, 4, 6, 4, 3, 4, 5, 8, 1, 4, 2, 9, 6

Offset: 1

Vaclav Kotesovec, Jun 13 2015

For m>0, Product_{k>=1} (1 + m/k^4) = (cosh(sqrt(2)*Pi*m^(1/4)) - cos(sqrt(2)*Pi*m^(1/4))) / (2*Pi^2*sqrt(m)). - Vaclav Kotesovec, Aug 30 2024

			2.16736062588226195190023136684702744182161317296349850975623259988...

Cf. A109219, A175615, A175617, A175619, A156648, A073017, A258871, A334411.

evalf((cosh(sqrt(2)*Pi)-cos(sqrt(2)*Pi))/(2*Pi^2),120);

RealDigits[(Cosh[Sqrt[2]*Pi]-Cos[Sqrt[2]*Pi])/(2*Pi^2),10,120][[1]]

exp(sumalt(j=1, -(-1)^j*zeta(4*j)/j)) \\ Vaclav Kotesovec, Dec 15 2020

Equals (cosh(sqrt(2)*Pi)-cos(sqrt(2)*Pi))/(2*Pi^2).

Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(4*j)/j)). - Vaclav Kotesovec, Mar 28 2019

A258871 Decimal expansion of Product_{n>=1} (1+1/n^6).

Original entry on oeis.org

2, 0, 3, 4, 7, 4, 0, 8, 3, 5, 0, 0, 9, 4, 2, 9, 0, 6, 3, 5, 8, 6, 8, 2, 0, 8, 0, 9, 6, 4, 2, 8, 5, 0, 8, 9, 7, 7, 1, 0, 9, 0, 1, 0, 0, 6, 2, 3, 9, 2, 5, 4, 6, 9, 0, 5, 5, 7, 5, 3, 9, 4, 8, 0, 4, 5, 2, 9, 8, 4, 1, 2, 0, 1, 9, 1, 5, 2, 5, 8, 4, 9, 1, 3, 5, 3, 5, 9, 8, 1, 5, 4, 9, 6, 6, 7, 0, 7, 6, 8, 6, 7, 8, 1, 3

Offset: 1

Vaclav Kotesovec, Jun 13 2015

From Vaclav Kotesovec, Aug 30 2024: (Start)
For m>0, Product_{k>=1} (1 + m/k^6) = (cosh(Pi*m^(1/6)) - cos(sqrt(3)*Pi*m^(1/6))) * sinh(Pi*m^(1/6)) / (2*Pi^3*sqrt(m)).
If m tends to infinity, Product_{k>=1} (1 + m/k^6) ~ exp(2*Pi*m^(1/6)) / (8*Pi^3*sqrt(m)). (End)

			2.03474083500942906358682080964285089771090100623925469055753948...

Cf. A109219, A175615, A175617, A175619, A156648, A073017, A258870, A334411.

evalf((cosh(Pi)-cos(sqrt(3)*Pi))*sinh(Pi)/(2*Pi^3), 120);

RealDigits[(Cosh[Pi]-Cos[Sqrt[3]*Pi])*Sinh[Pi]/(2*Pi^3),10,120][[1]]

prodnumrat(1+x^-6, 1) \\ Charles R Greathouse IV, Feb 04 2025

Equals (cosh(Pi)-cos(sqrt(3)*Pi))*sinh(Pi)/(2*Pi^3).

Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(6*j)/j)). - Vaclav Kotesovec, Mar 28 2019

A334411 Decimal expansion of Product_{k>=1} (1 + 1/k^8).

Original entry on oeis.org

2, 0, 0, 8, 1, 5, 6, 0, 5, 4, 9, 9, 2, 7, 4, 5, 3, 1, 5, 1, 4, 9, 0, 3, 9, 4, 8, 2, 3, 2, 3, 4, 1, 3, 6, 9, 2, 1, 1, 9, 5, 3, 2, 1, 5, 9, 8, 3, 0, 9, 5, 0, 9, 7, 8, 7, 7, 0, 7, 4, 2, 9, 9, 6, 1, 7, 4, 2, 2, 7, 2, 5, 1, 1, 3, 8, 0, 5, 5, 2, 0, 9, 3, 4, 0, 6, 0, 5, 0, 1, 0, 2, 0, 2, 6, 9, 6, 3, 0, 3, 2, 1, 8, 7, 9

Offset: 1

Vaclav Kotesovec, Apr 27 2020

From Vaclav Kotesovec, Aug 30 2024: (Start)
For m>0, Product_{k>=1} (1 + m/k^8) = (cosh(Pi*sqrt(2 - sqrt(2))*m^(1/8)) - cos(Pi*sqrt(2 + sqrt(2))*m^(1/8))) * (cosh(Pi*sqrt(2 + sqrt(2))*m^(1/8)) - cos(Pi*sqrt(2 - sqrt(2))*m^(1/8)))/(4*sqrt(m)*Pi^4).
If m tends to infinity, Product_{k>=1} (1 + m/k^8) ~ exp(Pi*sqrt(2*(2 + sqrt(2)))*m^(1/8)) / (16*Pi^4*sqrt(m)).
In general, if m tends to infinity and v > 2, Product_{k>=1} (1 + m/k^v) ~ exp(Pi*m^(1/v)/sin(Pi/v)) / ((2*Pi)^(v/2)*sqrt(m)). (End)

			2.00815605499274531514903948232341369211953215983095097877074299617422...

Cf. A156648, A073017, A258870, A307216, A258871.

Cf. A109219, A175615, A175616, A175617, A175619.

evalf(Product(1 + 1/j^8, j = 1..infinity), 120);

RealDigits[Chop[N[Product[(1 + 1/n^8), {n, 1, Infinity}], 120]]][[1]]

default(realprecision, 120); exp(sumalt(j=1, -(-1)^j*zeta(8*j)/j))

Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(8*j)/j)).

Equals (cos(sqrt(4 - 2*sqrt(2))*Pi) + cos(sqrt(4 + 2*sqrt(2))*Pi) + cosh(sqrt(4 - 2*sqrt(2))*Pi) + cosh(sqrt(4 + 2*sqrt(2))*Pi) - 2*cos(sqrt(2 - sqrt(2))*Pi) * cosh(sqrt(2 - sqrt(2))*Pi) - 2*cos(sqrt(2 + sqrt(2))*Pi) * cosh(sqrt(2 + sqrt(2))*Pi)) / (8*Pi^4).

A375840 a(n) = Product_{k=0..n} (k^3 + n).

Original entry on oeis.org

0, 2, 60, 3960, 505920, 111945600, 39501498960, 20891200176000, 15785674348953600, 16407441209402496000, 22748452701706791576000, 41018285140626186366336000, 94161166261926730618189824000, 270252010494895412092926136320000, 954766647796042233397162343121696000

Offset: 0

Vaclav Kotesovec, Aug 31 2024

nonn

Cf. A073017, A255433.

Cf. A375839, A375841, A375842, A375843, A375844, A375845.

Table[Product[k^3 + n, {k, 0, n}], {n, 0, 15}]

a(n) ~ n^(3*n + 2) / exp(3*n - 2*Pi*n^(1/3)/sqrt(3)).

A006014 a(n+1) = (n+1)a(n) + Sum_{k=1..n-1} a(k)a(n-k).

Original entry on oeis.org

1, 2, 7, 32, 178, 1160, 8653, 72704, 679798, 7005632, 78939430, 965988224, 12762344596, 181108102016, 2748049240573, 44405958742016, 761423731533286, 13809530704348160, 264141249701936818, 5314419112717217792, 112201740111374359516, 2480395343617443024896

Offset: 1

N. J. A. Sloane

			G.f. = x + 2*x^2 + 7*x^3 + 32*x^4 + 178*x^5 + 1160*x^6 + 8653*x^7 + 72704*x^8 + ...

D. E. Knuth, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael De Vlieger, Table of n, a(n) for n = 1..449
Jimmy Devillet and Bruno Teheux, Associative, idempotent, symmetric, and order-preserving operations on chains, arXiv:1805.11936 [math.RA], 2018.
E. Duchi, V. Guerrini, S. Rinaldi, and G. Schaeffer, Fighting fish. J. Phys. A, Math. Theor. 50, No. 2, Article ID 024002, 16 p. (2017), chapter 4.

Similar recurrences: A124758, A243499, A284005, A329369, A341392.

Cf. A007814, A025480, A073017.

Nest[Append[#1, #1[[-1]] (#2 + 1) + Total@ Table[#1[[k]] #1[[#2 - k]], {k, #2 - 1}]] & @@ {#, Length@ #} &, {1}, 17] (* Michael De Vlieger, Aug 22 2018 *)
(* or *)
a[1] = 1; a[n_] := a[n] = n a[n-1] + Sum[a[k] a[n-1-k], {k, n-2}]; Array[a, 18] (* Giovanni Resta, Aug 23 2018 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = k * A[k-1] + sum( j=1, k-2, A[j] * A[k-1-j])); A[n])} /* Michael Somos, Jul 24 2011 */

from sage.all import *
@CachedFunction
def a(n): # a = A006014
    if n<5: return (pow(5,n-1) + 3)//4
    else: return n*a(n-1) + 2*sum(a(k)*a(n-k-1) for k in range(1,(n//2))) + (n%2)*pow(a((n-1)//2),2)
print([a(n) for n in range(1,61)]) # G. C. Greubel, Jan 10 2025

G.f. A(x) satisfies A(x) = x * (1 + A(x) + A(x)^2 + x * A'(x)). - Michael Somos, Jul 24 2011
Conjecture: a(n) = Sum_{k=0..2^(n-1) - 1} b(k) for n > 0 where b(2n+1) = b(n), b(2n) = b(n) + b(n - 2^f(n)) + b(2n - 2^f(n)) + b(A025480(n-1)) for n > 0 with b(0) = b(1) = 1 and where f(n) = A007814(n). - Mikhail Kurkov, Nov 19 2021
a(n) ~ cosh(sqrt(3)*Pi/2) * n! / Pi = A073017 * n!. - Vaclav Kotesovec, Nov 21 2024

A307209 Decimal expansion of Product_{i>=1, j>=1} (1 + 1/(i^3 + j^3)).

Original entry on oeis.org

3, 5, 0, 4, 7, 8, 2, 9, 9, 9, 3, 3, 9, 7, 2, 8, 3, 7, 5, 8, 9, 1, 1, 2, 0, 5, 7, 0, 4, 3, 8, 0, 6, 1, 2, 5, 5, 8, 3, 8, 9, 3, 2, 4, 7, 8, 6, 2, 7, 1, 2, 7, 5, 3, 5, 4, 1, 9, 9, 4, 6, 2, 6, 6, 1, 4, 0, 5, 8, 3, 8, 5, 0, 3, 5, 0, 3, 4, 7, 5, 6, 3, 5, 2, 7, 4, 7, 5, 0, 9, 5, 0, 5, 1, 3, 7, 8, 9, 1, 7, 8, 4, 5, 9, 7

Offset: 1

Vaclav Kotesovec, Mar 28 2019

Product_{i>=1, j>=1} (1 + 1/(i^2 + j^2)) is divergent.
A324443(n) / A324403(n) ~ c * n^(Pi/2), where c = A306398 * 2^(3/4) * exp(-Pi/12) * Pi^(1/4) * Gamma(3/4) = 0.36753062884677326134620846786416595535234038999313...
Product_{i=1..n, j=1..n} (1 + 1/(i + j)) = A324444(n) / A079478(n) ~ 2^(2*n + 1) / (sqrt(Pi)*n^(3/2)).

			3.50478299933972837589112057043806125583893247862712753541994626614058385...

Vaclav Kotesovec, Table of n, a(n) for n = 1..224

Cf. A073017, A307210, A307215, A324403, A324426, A324443.

(* The iteration cycle: *) $MaxExtraPrecision = 1000; funs[n_] := Product[1 + 1/(i^3 + j^3), {i, 1, n}, {j, 1, n}]; Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[200/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 100]], {m, 10, 100, 10}]

default(realprecision, 50); exp(sumalt(k=1, -(-1)^k/k*sumnum(i=1, sumnum(j=1, 1/(i^3+j^3)^k)))) \\ 15 decimals correct

Equals limit_{n->infinity} A307210(n) / A324426(n).

A326865 G.f.: Product_{k>=1} (1 + x^k/k^3) = Sum_{n>=0} a(n)*x^n/n!^3.

Original entry on oeis.org

1, 1, 1, 35, 728, 48824, 7170984, 1418111064, 479963197440, 235727037775872, 170423013422592000, 163854260184125952000, 214343327259234349056000, 360795240553638133592064000, 778954481701636984110452736000, 2095759092922096320907078496256000

Offset: 0

Vaclav Kotesovec, Jul 27 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..182

Cf. A007838, A249593, A326864.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 27 2023

nmax = 20; CoefficientList[Series[Product[(1+x^k/k^3), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!^3

a(n) ~ c * (n-1)!^3, where c = A073017 = Product_{k>=1} (1 + 1/k^3) = cosh(sqrt(3)*Pi/2)/Pi = 2.428189792098870328736...

cp's OEIS Frontend

A045778 Number of factorizations of n into distinct factors greater than 1.

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A156648 Decimal expansion of Product_{k>=1} (1 + 1/k^2).

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PARI

PARI

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A109219 Decimal expansion of Product_{n >= 2} 1-n^(-3).

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Mathematica

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PARI

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A258870 Decimal expansion of Product_{n>=1} (1+1/n^4).

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A258871 Decimal expansion of Product_{n>=1} (1+1/n^6).

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Maple

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PARI

Formula

A334411 Decimal expansion of Product_{k>=1} (1 + 1/k^8).

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A006014 a(n+1) = (n+1)a(n) + Sum_{k=1..n-1} a(k)a(n-k).