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A007425
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d_3(n), or tau_3(n), the number of ordered factorizations of n as n = r s t.
(Formerly M2282)
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136
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1, 3, 3, 6, 3, 9, 3, 10, 6, 9, 3, 18, 3, 9, 9, 15, 3, 18, 3, 18, 9, 9, 3, 30, 6, 9, 10, 18, 3, 27, 3, 21, 9, 9, 9, 36, 3, 9, 9, 30, 3, 27, 3, 18, 18, 9, 3, 45, 6, 18, 9, 18, 3, 30, 9, 30, 9, 9, 3, 54, 3, 9, 18, 28, 9, 27, 3, 18, 9, 27, 3, 60, 3, 9, 18, 18, 9, 27, 3, 45, 15, 9, 3, 54, 9, 9, 9, 30, 3
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OFFSET
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1,2
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COMMENTS
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Let n = Product p_i^e_i. Tau (A000005) is tau_2, this sequence is tau_3, A007426 is tau_4, where tau_k(n) (also written as d_k(n)) = Product_i binomial(k-1+e_i, k-1) is the k-th Piltz function. It gives the number of ordered factorizations of n as a product of k terms. - Len Smiley
Inverse Möbius transform applied twice to all 1's sequence.
Appears to equal the number of plane partitions of n that can be extended in exactly 3 ways to a plane partition of n+1 by adding one element. - Wouter Meeussen, Sep 11 2004
Number of plane partitions of n that can be extended in exactly 3 ways to a plane partition of n+1 by adding one element. If the partition is not a box, there is a minimal i+j where b_{i,j} != b_{1,1} and an element can be added there. - Franklin T. Adams-Watters, Jun 14 2006
a(n) is congruent to 1 (mod 3) if n is a perfect cube, otherwise a(n) is congruent to 0 (mod 3). - Geoffrey Critzer, Mar 20 2015
Number of 3D grids of n congruent boxes with three different edge lengths, in a box, modulo rotation (cf. A034836 for cubes instead of boxes and A140773 for boxes with two different edge lengths; cf. A000005 for the 2D case). - Manfred Boergens, Apr 06 2021
Number of ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022
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REFERENCES
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M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.
A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.
Paul J. McCarthy, Introduction to Arithmetical Functions, Springer, 1986.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Karin Cvetko-Vah, Michael Kinyon, Jonathan Leech, and Tomaž Pisanski, Regular Antilattices, arXiv:1911.02858 [math.RA], 2019.
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FORMULA
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Dirichlet g.f.: zeta^3(s).
a(n^2) = tau_3(n^2) = tau_2(n^2)*tau_2(n), where tau_2 is A000005 and tau_3 is this sequence.
a(s) = 3^omega(s), if s>1 is squarefree (A005117) and omega(s) is: A001221. (End)
a(n) = tau_3(n) = tau_2(n)*tau_2(n*rad(n))/tau_2(rad(n)), where rad(n) is A007947 and tau_2(n) is A000005.
tau_3(n) >= 2*tau_2(n) - 1.
tau_3(n) <= tau_2(n)^2 + tau_2(n)-1. (End)
a(n) = sqrt(Sum_{d|n}(tau(d))^3);
a(n) = |Sum_{d|n} A008836(d)*(tau(d))^2)|.
The first formula follows from the first Cloitre formula and a Liouville formula; the second formula follows from our analogous formula (cf. our comment in Formula section of A000005). (End)
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(tau(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
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EXAMPLE
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a(6) = 9; the divisors of 6 are {1,2,3,6} and the numbers of divisors of these divisors are 1, 2, 2, and 4. Adding them, we get 9 as a result.
Also, since 6 is a squarefree number, the formula from Herrero can be used to obtain the result: a(6) = 3^omega(6) = 3^2 = 9. - Wesley Ivan Hurt, May 30 2014
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MAPLE
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f:=proc(n) local t1, i, j, k; t1:=0; for i from 1 to n do for j from 1 to n do for k from 1 to n do if i*j*k = n then t1:=t1+1; fi; od: od: od: t1; end;
A007425 := proc(n) local e, j; e := ifactors(n)[2]: product(binomial(2+e[j][2], 2), j=1..nops(e)); end; # Len Smiley
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MATHEMATICA
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f[n_] := Plus @@ DivisorSigma[0, Divisors[n]]; Table[ f[n], {n, 90}] (* Robert G. Wilson v, Sep 13 2004 *)
SetAttributes[tau, Listable]; tau[1, n_] := 1; tau[k_, n_] := Plus @@ (tau[k-1, Divisors[n]]); Table[tau[3, n], {n, 100}] (* Enrique Pérez Herrero, Nov 08 2009 *)
Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 50}] (* Wesley Ivan Hurt, May 30 2014 *)
f[p_, e_] := (e+1)*(e+2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 27 2019 *)
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PROG
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(PARI) for(n=1, 100, print1(sumdiv(n, k, numdiv(k)), ", "))
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)^3)[n]) \\ Ralf Stephan
(PARI) a(n)=sumdiv(n, x, sumdiv(x, y, 1 )) \\ Joerg Arndt, Oct 07 2012
(Haskell)
a007425 = sum . map a000005 . a027750_row
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CROSSREFS
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KEYWORD
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nonn,nice,easy,mult
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AUTHOR
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STATUS
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approved
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