A134577 -id:A134577 - cp's OEIS frontend

A007425 d_3(n), or tau_3(n), the number of ordered factorizations of n as n = r s t.

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

Offset: 1

N. J. A. Sloane, May 24 1994

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.
A085782 gives the range of values of this sequence. - Matthew Vandermast, Jul 12 2004
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 divisors of n's divisors. - Lekraj Beedassy, Sep 07 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
Equals row sums of A127170. - Gary W. Adamson, May 20 2007
Equals A134577 * [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, Nov 02 2007
Equals row sums of triangle A143354. - Gary W. Adamson, Aug 10 2008
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
Also row sums of A195050. - Omar E. Pol, Nov 26 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

			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

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).

T. D. Noe, Table of n, a(n) for n = 1..10000
R. Eisinga, R. Breitling, and T. Heskes, The exact probability distribution of the rank product statistics for replicated experiments, FEBS Letters, 2013, 587: 677-682, doi:10.1016/j.febslet.2013.01.037
Karin Cvetko-Vah, Michael Kinyon, Jonathan Leech, and Tomaž Pisanski, Regular Antilattices, arXiv:1911.02858 [math.RA], 2019.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
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.
N. J. A. Sloane, Transforms.
Qingfeng Sun and Deyu Zhang, Sums of the triple divisor function over values of a ternary quadratic form, arXiv:1510.06170 [math.NT], 2015.
E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics 1 (1942): 129-152.
Wikipedia, Adolf Piltz.

Cf. A000005 (Mobius transform), A007426 (inverse Mobius transform), A061201 (partial sums), A127270, A143354, A027750, A007428 (Dirichlet inverse), A175596.
Column k=3 of A077592.
Additional cross-references mentioned in a comment: A034836, A038548, A140733.

a007425 = sum . map a000005 . a027750_row
-- Reinhard Zumkeller, Feb 16 2012

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

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 *)

for(n=1,100,print1(sumdiv(n,k,numdiv(k)),","))

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^3)[n]) \\ Ralf Stephan

a(n)=sumdiv(n, x, sumdiv(x, y, 1 )) \\ Joerg Arndt, Oct 07 2012

a(n)=sumdivmult(n,k,numdiv(k)) \\ Charles R Greathouse IV, Aug 30 2013

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^3)[n]), ", ")) \\ Vaclav Kotesovec, May 06 2025

from math import prod, comb
from sympy import factorint
def A007425(n): return prod(comb(2+e,2) for e in factorint(n).values()) # Chai Wah Wu, Dec 22 2024

a(n) = Sum_{d dividing n} tau(d). - Benoit Cloitre, Apr 04 2002
G.f.: Sum_{k>=1} tau(k)*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
For n = Product p_i^e_i, a(n) = Product_i A000217(e_i + 1). - Lekraj Beedassy, Sep 07 2004
Dirichlet g.f.: zeta^3(s).
From Enrique Pérez Herrero, Nov 03 2009: (Start)
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)
From Enrique Pérez Herrero, Nov 08 2009: (Start)
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)
From Vladimir Shevelev, Dec 22 2017: (Start)
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

A007429 Inverse Moebius transform applied twice to natural numbers.

Original entry on oeis.org

1, 4, 5, 11, 7, 20, 9, 26, 18, 28, 13, 55, 15, 36, 35, 57, 19, 72, 21, 77, 45, 52, 25, 130, 38, 60, 58, 99, 31, 140, 33, 120, 65, 76, 63, 198, 39, 84, 75, 182, 43, 180, 45, 143, 126, 100, 49, 285, 66, 152, 95, 165, 55, 232, 91, 234, 105, 124, 61, 385, 63, 132, 162, 247, 105, 260

Offset: 1

N. J. A. Sloane

Sum of the divisors d1 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

a(n) is the sum of the sum-of-divisors of the divisors of n. - M. F. Hasler, Mar 29 2024

David M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Travis Scholl, Table of n, a(n) for n = 1..100000 (terms 1 through 1000 were by T. D. Noe)
Olivier Bordelles, Mean values of generalized gcd-sum and lcm-sum functions, JIS, Vol. 10 (2007), Article 07.9.2, series g_4 (with an apparently wrong D.g.f. after equation 3).
N. J. A. Sloane, Transforms.

Cf. A000203, A007430, A134699, A152649, A280077.

[&+[SumOfDivisors(d): d in Divisors(n)]: n in [1..100]] // Jaroslav Krizek, Sep 24 2016

A007429 := proc(n)
    add(numtheory[sigma](d),d=numtheory[divisors](n)) ;
end proc:
seq(A007429(n),n=1..100) ; # R. J. Mathar, Aug 28 2015

f[n_] := Plus @@ DivisorSigma[1, Divisors@n]; Array[f, 52] (* Robert G. Wilson v, May 05 2010 *)
f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 10 2022 *)

A007429_upto(N)=vector(N,n, sumdiv(n,d, sigma(d))) \\ edited by M. F. Hasler, Mar 29 2024

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^2/(1-p*X))[n]) \\ Ralf Stephan

N=17; default(seriesprecision,N); x=z+O(z^(N+1))
c=sum(j=1,N,j*x^j);
t=1/prod(j=1,N, eta(x^(j))^(1/j))
t=log(t)
t=serconvol(t,c)
Vec(t)
/* Joerg Arndt, May 03 2008 */

a(n)=sumdiv(n,d, sumdiv(d,t, t ) );  /* Joerg Arndt, Oct 07 2012 */

from math import prod
from sympy import factorint
def A007429(n): return prod((p*(p**(e+1)-1)-(p-1)*(e+1))//(p-1)**2 for p,e in factorint(n).items()) # Chai Wah Wu, Mar 28 2024

def A(n): return sum(sigma(d) for d in n.divisors()) # Travis Scholl, Apr 14 2016

a(n) = Sum_{d|n} sigma(d), Dirichlet convolution of A000203 and A000012. - Jason Earls, Jul 07 2001
a(n) = Sum_{d|n} d*tau(n/d). - Vladeta Jovovic, Jul 31 2002
Multiplicative with a(p^e) = (p*(p^(e+1)-1)-(p-1)*(e+1))/(p-1)^2. - Vladeta Jovovic, Dec 25 2001
G.f.: Sum_{k>=1} sigma(k)*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Moebius transform of A007430. - Benoit Cloitre, Mar 03 2004
Dirichlet g.f.: zeta(s-1)*zeta^2(s).
Equals A051731^2 * [1, 2, 3, ...]. Equals row sums of triangle A134577. - Gary W. Adamson, Nov 02 2007
Row sums of triangle A134699. - Gary W. Adamson, Nov 06 2007
a(n) = n * (Sum_{d|n} tau(d)/d) = n * (A276736(n) / A276737(n)). - Jaroslav Krizek, Sep 24 2016
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(sigma(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 26 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). - Amiram Eldar, Oct 22 2022

A007433 Inverse Moebius transform applied twice to squares.

Original entry on oeis.org

1, 6, 11, 27, 27, 66, 51, 112, 102, 162, 123, 297, 171, 306, 297, 453, 291, 612, 363, 729, 561, 738, 531, 1232, 678, 1026, 922, 1377, 843, 1782, 963, 1818, 1353, 1746, 1377, 2754, 1371, 2178, 1881, 3024, 1683

Offset: 1

N. J. A. Sloane

Dirichlet convolution of A001157 and A000012. Dirichlet convolution of A000005 and A000290 (Jovovic formula). - R. J. Mathar, Feb 03 2011

Sum of the squares of the divisors d1 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

			G.f. = x + 6*x^2 + 11*x^3 + 27*x^4 + 27*x^5 + 66*x^6 + 51*x^7 + 112*x^8 + 102*x^9 + ... - _Michael Somos_, Jul 15 2018

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms

Cf. A134577.

a[n_] := Plus @@ DivisorSigma[2, Divisors[n]]; Array[a, 41] (* Robert G. Wilson v, May 05 2010 *)
a[ n_] := If[ n < 1, 0, Times @@ (If[ # == 1, 1, (#^(2 #2 + 4) - (#2 + 2) #^2 + #2 + 1) / (#^2 - 1)^2] & @@@ FactorInteger @ n)]; (* Michael Somos, Jul 15 2018 *)

/* Dirichlet convolution of A001157, A000012 (Mathar): */
a(n)=sumdiv(n, d, sigma(d,2))

/* Dirichlet convolution of A000005, A000290 (Mathar): */
a(n)=sumdiv(n, d, d^2*sigma(n/d,0))

a(n) = Sum_{d|n} d^2*tau(n/d). - Vladeta Jovovic, Jul 31 2002
Equals A134577 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 02 2007
G.f.: Sum_{k>=1} sigma_2(k)*x^k/(1 - x^k), where sigma_2(k) is the sum of squares of divisors of k (A001157). - Ilya Gutkovskiy, Jan 16 2017
Dirichlet g.f.: zeta(s-2)*zeta(s)^2. - Benedict W. J. Irwin, Jul 14 2018
a(n) is multiplicative with a(p^e) = (p^(2*e + 4) - (e+2) * p^2 + e+1) / (p^2 - 1)^2. - Michael Somos, Jul 15 2018
Sum_{k=1..n} a(k) ~ Zeta(3)^2 * n^3 / 3. - Vaclav Kotesovec, Nov 04 2018

a(38) corrected by Ilya Gutkovskiy, Jan 16 2016

A034761 Dirichlet convolution of sigma(n) with itself.

Original entry on oeis.org

1, 6, 8, 23, 12, 48, 16, 72, 42, 72, 24, 184, 28, 96, 96, 201, 36, 252, 40, 276, 128, 144, 48, 576, 98, 168, 184, 368, 60, 576, 64, 522, 192, 216, 192, 966, 76, 240, 224, 864, 84, 768, 88, 552, 504, 288, 96, 1608, 178, 588, 288, 644, 108, 1104, 288, 1152, 320, 360

Offset: 1

Erich Friedman

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A000005, A000203 (sigma), A001620, A073002, A074962, A134577.

f[p_, e_] := ((e + 1)*p^(e + 3) - (e + 3)*(p^(e + 2) - p + 1) + 2)/(p - 1)^3; f[2, e_] := (e - 1)*2^(e + 2) + e + 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 16 2022 *)

Dirichlet g.f.: zeta^2(s)*zeta^2(s-1).
Multiplicative with a(2^e) = (e-1) 2^(e+2) + e + 5, a(p^e) = ((1+e)p^(e+3) - (3+e)(p^(e+2)-p+1) + 2)/(p-1)^3, p > 2. - Mitch Harris, Jun 27 2005 [corrected by Amiram Eldar, Oct 16 2022 and Sep 12 2023]
Equals A134577 * A000005. - Gary W. Adamson, Nov 02 2007
Also the Dirichlet convolution A000005 by A038040. - R. J. Mathar, Apr 01 2011
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 * (2*Pi^2 * log(n) + (4*gamma - 1)*Pi^2 + 24*zeta'(2)) / 144, where gamma is the Euler-Mascheroni constant A001620 and Zeta'(2) = A073002. Equivalently, Sum_{k=1..n} a(k) ~ Pi^4 * n^2 * (2*log(n) - 1 + 8*gamma - 48*log(A) + 4*log(2*Pi)) / 144, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 28 2019

A329323 Triangle read by rows: T(n,k) is the sum of the parts congruent to 0 mod k in the partitions of n into equal parts, 1 <= k <= n.

Original entry on oeis.org

1, 4, 2, 6, 0, 3, 12, 8, 0, 4, 10, 0, 0, 0, 5, 24, 12, 12, 0, 0, 6, 14, 0, 0, 0, 0, 0, 7, 32, 24, 0, 16, 0, 0, 0, 8, 27, 0, 18, 0, 0, 0, 0, 0, 9, 40, 20, 0, 0, 20, 0, 0, 0, 0, 10, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 72, 48, 36, 24, 0, 24, 0, 0, 0, 0, 0, 12, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 56, 28, 0, 0

Offset: 1

Omar E. Pol, Nov 21 2019

Column k lists the terms of A038040 multiplied by k and interspersed with (k-1) zeros.

			Triangle begins:
   1;
   4,  2;
   6,  0,  3;
  12,  8,  0,  4;
  10,  0,  0,  0,  5;
  24, 12, 12,  0,  0,  6;
  14,  0,  0,  0,  0,  0,  7;
  32, 24,  0, 16,  0,  0,  0,  8;
  27,  0, 18,  0,  0,  0,  0,  0,  9;
  40, 20,  0,  0, 20,  0,  0,  0,  0, 10;
  22,  0,  0,  0,  0,  0,  0,  0,  0,  0, 11;
  72, 48, 36, 24,  0, 24,  0,  0,  0,  0,  0, 12;
  26,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 13;
  56, 28,  0,  0,  0,  0, 28,  0,  0,  0,  0,  0,  0, 14;
  60,  0, 30,  0, 30,  0,  0,  0,  0,  0,  0,  0,  0,  0, 15;
  80, 64,  0, 48,  0,  0,  0, 32,  0,  0,  0,  0,  0,  0,  0, 16;
...
For n = 6 the partitions of 6 into equal parts are [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. Then, for k = 2 the sum of the parts that are multiples of 2 is 6 + 2 + 2 + 2 = 12, so T(6,2) = 12.

Column 1 is A038040.
Row sums give A034718.
Leading diagonal gives A000027.
The number of positive terms in row n is A000005(n).
Cf. A126988, A130540, A134577, A244051.

T(n,k) = A126988(n,k)*A134577(n,k).

cp's OEIS Frontend

A007425 d_3(n), or tau_3(n), the number of ordered factorizations of n as n = r s t.

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A007429 Inverse Moebius transform applied twice to natural numbers.

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A007433 Inverse Moebius transform applied twice to squares.

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A034761 Dirichlet convolution of sigma(n) with itself.

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A329323 Triangle read by rows: T(n,k) is the sum of the parts congruent to 0 mod k in the partitions of n into equal parts, 1 <= k <= n.

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