A007425 -id:A007425 - cp's OEIS frontend

A061201 Partial sums of A007425: (tau<=)_3(n).

1, 4, 7, 13, 16, 25, 28, 38, 44, 53, 56, 74, 77, 86, 95, 110, 113, 131, 134, 152, 161, 170, 173, 203, 209, 218, 228, 246, 249, 276, 279, 300, 309, 318, 327, 363, 366, 375, 384, 414, 417, 444, 447, 465, 483, 492, 495, 540, 546, 564, 573, 591, 594, 624, 633, 663

Offset: 1

Vladeta Jovovic, Apr 21 2001

(tau<=)_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k<=n}|, i.e., tau<=_k(n) is number of solutions to x_1*x_2*...*x_k<=n, x_i > 0.
A061201(n) is the number of 4-tuples (w,x,y,z) having all terms in {1,...,n} and w=x*y*z; see A211795 for a list of related counting sequences. - Clark Kimberling, Apr 28 2012
The formula for Sum_{k=1..n} d3(k) in the Benoit Cloitre article on page 15 is incorrect. For correct asymptotic formula see below or generate it in the Mathematica: Residue[Zeta[s]^3 * n^s/s, {s, 1}] // Expand. - Vaclav Kotesovec, Aug 19 2021

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000 (terms 1..1000 from Harry J. Smith)
Benoit Cloitre, On the divisor and circle problems
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_5(n): A061200, tau_6(n): A034695, (unordered) 2-factorizations of n: A038548, (unordered) 3-factorizations of n: A034836, A001055, (tau<=)_2(n): A006218, (tau<=)_4(n): A061202, (tau<=)_5(n): A061203, (tau<=)_6(n): A061204.

[&+[NumberOfDivisors(k)*Floor(n/k): k in [1..n]]: n in [1..56]];  // Bruno Berselli, Apr 13 2011

b:= proc(k, n) option remember; uses numtheory;
     `if`(k=1, 1, add(b(k-1, d), d=divisors(n)))
    end:
a:= proc(n) option remember; `if`(n=0, 0, b(3, n)+a(n-1)) end:
seq(a(n), n=1..76);  # Alois P. Heinz, Oct 23 2023

a[n_] := Sum[ DivisorSigma[0, k]*Floor[n/k], {k, 1, n}]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Sep 20 2011, after Benoit Cloitre *)
(* Asymptotics: *) n*(Log[n]^2/2 + (3*EulerGamma - 1)*Log[n] + 3*EulerGamma^2 - 3*EulerGamma - 3*StieltjesGamma[1] + 1) (* Vaclav Kotesovec, Sep 09 2018 *)
Accumulate[a[n_]:=DivisorSum[n, DivisorSigma[0, #]&]; Array[a, 60]] (* Vincenzo Librandi, Jan 12 2020 *)

a(n)=sum(k=1,n,numdiv(k)*floor(n/k)) \\ Benoit Cloitre, Apr 19 2007

{ for (n=1, 1000, write("b061201.txt", n, " ", sum(k=1, n, numdiv(k)*(n\k))) ) } \\ Harry J. Smith, Jul 18 2009

my(N=60, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k)*x^k/(1-x^k))/(1-x)) \\ Seiichi Manyama, Jul 24 2022

from math import isqrt
from sympy import integer_nthroot
def A061201(n): return (m:=integer_nthroot(n,3)[0])**3+3*sum(-(s:=isqrt(r:=n//i))**2+(sum(r//k for k in range(1,s+1))<<1)-sum(n//(i*j) for j in range(1,m+1)) for i in range(1,m+1)) # Chai Wah Wu, Oct 23 2023

(tau<=)k(n) = Sum{i=1..n} tau_k(i).
a(n) = n * ( log(n)^2/2 + (3*g-1)*log(n) + 3*g^2-3*g-3*g1+1 ) + O(sqrt(n)), where g is the Euler-Mascheroni number ~ 0.57721... (see A001620), and g1 is the first Stieltjes constant ~ -0.072816 (see A082633). The determination of the precise size of the error term is an unsolved problem - see references. - Andrew Lelechenko, Apr 15 2011 [corrected by Vaclav Kotesovec, Sep 09 2018]
a(n) = Sum_{k=1..n} A000005(k)*floor(n/k). - Benoit Cloitre, Apr 19 2007
To compute a(n) for huge n (see A180365) in sublinear use a(n) = 3*Sum_{i=1..n3} A006218(n/i) - Sum_{j=1..n3} floor(n/(i*j)) + n3^3, where n3 = floor(n^(1/3)). - Andrew Lelechenko, Apr 15 2011
a(n) = Sum_{k=1..n} Sum_{i=1..n} floor(n/(i*k)). - Wesley Ivan Hurt, Sep 14 2017
G.f.: (1/(1-x)) * Sum_{k>=1} A000005(k) * x^k/(1 - x^k). - Seiichi Manyama, Jul 24 2022

A077592 Table by antidiagonals of tau_k(n), the k-th Piltz function (see A007425), or n-th term of the sequence resulting from applying the inverse Möbius transform (k-1) times to the all-ones sequence.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 9, 2, 1, 1, 8, 7, 21, 5, 16, 3, 4, 1, 1, 9, 8, 28, 6, 25, 4, 10, 3, 1, 1, 10, 9, 36, 7, 36, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 49, 6, 35, 10, 9, 2, 1, 1, 12, 11, 55, 9, 64, 7, 56, 15, 16, 3, 6, 1

Offset: 1

Henry Bottomley, Nov 08 2002

As an array with offset n=0, k=1, also the number of length n chains of divisors of k. - Gus Wiseman, Aug 04 2022

			T(6,3) = 9 because we have: 1*1*6, 1*2*3, 1*3*2, 1*6*1, 2*1*3, 2*3*1, 3*1*2, 3*2*1, 6*1*1. - _Geoffrey Critzer_, Feb 16 2015
From _Gus Wiseman_, May 03 2021: (Start)
Array begins:
       k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=0:  1   1   1   1   1   1   1   1
  n=1:  1   2   2   3   2   4   2   4
  n=2:  1   3   3   6   3   9   3  10
  n=3:  1   4   4  10   4  16   4  20
  n=4:  1   5   5  15   5  25   5  35
  n=5:  1   6   6  21   6  36   6  56
  n=6:  1   7   7  28   7  49   7  84
  n=7:  1   8   8  36   8  64   8 120
  n=8:  1   9   9  45   9  81   9 165
The triangular form T(n,k) = A(n-k,k) gives the number of length n - k chains of divisors of k. It begins:
  1
  1  1
  1  2  1
  1  3  2  1
  1  4  3  3  1
  1  5  4  6  2  1
  1  6  5 10  3  4  1
  1  7  6 15  4  9  2  1
  1  8  7 21  5 16  3  4  1
  1  9  8 28  6 25  4 10  3  1
  1 10  9 36  7 36  5 20  6  4  1
  1 11 10 45  8 49  6 35 10  9  2  1
(End)

Alois P. Heinz, Antidiagonals n = 1..141, flattened
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.
Wikipedia, Adolf Piltz.

Rows include: A000012, A000005, A034695, A111217, A111218, A111219, A111220, A111221, A111306.
Columns include (with multiplicity and some offsets) A000012, A000027, A000027, A000217, A000027, A000290, A000027, A000292, A000217, A000290, A000027, A002411, A000027, A000290, A000290, A000332 etc.
Cf. A077593.
Row n = 2 of the array is A007425.
Row n = 3 of the array is A007426.
Row n = 4 of the array is A061200.
The diagonal n = k of the array (central column of the triangle) is A163767.
The transpose of the array is A334997.
Diagonal n = k of the array is A343939.
Antidiagonal sums of the array (or row sums of the triangle) are A343940.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
A251683(n,k) counts strict length k + 1 chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict length k chains of divisors from n to 1.
A337255(n,k) counts strict length k chains of divisors starting with n.
Cf. A018892, A051026, A062319, A143773, A176029, A327527, A337256, A343656, A343658, A343662.

with(numtheory):
A:= proc(n,k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=divisors(n)))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..14);  # Alois P. Heinz, Feb 25 2015

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[tau[n - k + 1, k], {n, 14}, {k, n, 1, -1}] // Flatten (* Robert G. Wilson v *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#] + k - 1, k - 1] & /@ FactorInteger[n]); Table[tau[k, n - k + 1], {n, 1, 13}, {k, 1, n}] // Flatten (* Amiram Eldar, Sep 13 2020 *)
Table[Length[Select[Tuples[Divisors[k],n-k],And@@Divisible@@@Partition[#,2,1]&]],{n,12},{k,1,n}] (* TRIANGLE, Gus Wiseman, May 03 2021 *)
Table[Length[Select[Tuples[Divisors[k],n-1],And@@Divisible@@@Partition[#,2,1]&]],{n,6},{k,6}] (* ARRAY, Gus Wiseman, May 03 2021 *)

If n = Product_i p_i^e_i, then T(n,k) = Product_i C(k+e_i-1, e_i). T(n,k) = Sum_d{d|n} T(n-1,d) = A077593(n,k) - A077593(n-1,k).
Columns are multiplicative.
Dirichlet g.f. for column k: Zeta(s)^k. - Geoffrey Critzer, Feb 16 2015
A(n,k) = A334997(k,n). - Gus Wiseman, Aug 04 2022

Typo in formula fixed by Geoffrey Critzer, Feb 16 2015

A180365 Partial sums up to 10^n of A007425.

Original entry on oeis.org

1, 53, 1471, 29425, 496623, 7518850, 106030594, 1421760251, 18362473634, 230375375227, 2824280446479, 33978264556380, 402439152166882, 4703975577244852, 54365786164534232, 622220063695532731, 7060841638683030355, 79525016215415440837

Offset: 0

Andrew Lelechenko, Jan 19 2011

nonn

Henri Lifchitz, Table of n, a(n) for n = 0..23 (first 21 terms from Hiroaki Yamanouchi)

Cf. A007425, A061201.

a(n)=sum(k=1, 10^n, numdiv(k)*floor((10^n)/k));
vector(7, n, a(n-1)) \\ Altug Alkan, Sep 24 2015

from math import isqrt
from sympy import integer_nthroot
def A180365(n): return (m:=integer_nthroot(t:=10**n,3)[0])**3+3*sum(-(s:=isqrt(r:=t//i))**2+(sum(r//k for k in range(1,s+1))<<1)-sum(t//(i*j) for j in range(1,m+1)) for i in range(1,m+1)) # Chai Wah Wu, Oct 23 2023

a(n) = A061201(10^n).

One more term (a(16)) and typo in crossrefs fixed by Andrew Lelechenko, Apr 13 2011

a(17) from Hiroaki Yamanouchi, Jul 15 2014

A308042 Decimal expansion of the asymptotic mean of d_3(k)/ud_3(k), where d_3(k) is the number of ordered factorizations of k as product of 3 divisors (A007425) and ud_3(k) = 3^omega(k) is the unitary analog of d_3 (A074816).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 10 2019

			2.22416248380186958442174889454690037857600080851428...

Meselem Karras and Abdallah Derbal, Mean value of an arithmetic function associated with the Piltz divisor function, Asian-European Journal of Mathematics, Vol. 13, No. 3 (2018), 2050062.

Cf. A001221, A007425, A074816, A307869.

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{6, -16, 64/3, -32/3}, {0, 8, 32, 224/3}, m]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n/2^n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

prodeulerrat((1 - 1/p) * (2 + (1 - 1/p)^(-3))/3) \\ Amiram Eldar, Sep 16 2024

Equals Product_{p prime} ((1 - 1/p) * (2 + (1 - 1/p)^(-3))/3).

A331080 a(n) = Sum_{i=1..n} d_3(i)*d_3(i+1), where d_3(n) = A007425(n).

Original entry on oeis.org

3, 12, 30, 48, 75, 102, 132, 192, 246, 273, 327, 381, 408, 489, 624, 669, 723, 777, 831, 993, 1074, 1101, 1191, 1371, 1425, 1515, 1695, 1749, 1830, 1911, 1974, 2163, 2244, 2325, 2649, 2757, 2784, 2865, 3135, 3225, 3306, 3387, 3441, 3765, 3927, 3954, 4089, 4359, 4467, 4629, 4791, 4845, 4935, 5205, 5475

Offset: 1

N. J. A. Sloane, Jan 10 2020

nonn

For background references see A330570.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A007425, A331081, A331130.

f[p_, e_] := (e+1)*(e+2)/2;  s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{t = Array[s, 100]}, Accumulate[Most[t] * Rest[t]]] (* Amiram Eldar, Apr 19 2024 *)

a(n) = 3 * A331081(n). - Amiram Eldar, Apr 19 2024

A175596 Partial products of A007425.

Original entry on oeis.org

1, 3, 9, 54, 162, 1458, 4374, 43740, 262440, 2361960, 7085880, 127545840, 382637520, 3443737680, 30993639120, 464904586800, 1394713760400, 25104847687200, 75314543061600, 1355661775108800, 12200955975979200, 109808603783812800, 329425811351438400, 9882774340543152000, 59296646043258912000, 533669814389330208000, 5336698143893302080000, 96060566590079437440000, 288181699770238312320000, 7780905893796434432640000

Offset: 1

Jonathan Vos Post, Dec 03 2010

nonn

Partial products of the number of ordered factorizations of n as a product of 3 terms.

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = d_4(gcd(i,j)) for 1 <= i,j <= n, where d_4(n) = A007426(n). - Enrique Pérez Herrero, Jan 20 2013

			a(8) = 1 * 3 * 3 * 6 * 3 * 9 * 3 * 10 = 43740 = 2^2 * 3^7 * 5.

Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.

Cf. A000005, A007425, A007426, A061201 (partial sums), A127270, A143354.

Cf. A066843.

Table[Product[Sum[DivisorSigma[0, d], {d, Divisors[k]}], {k, 1, n}], {n, 1, 30}] (* Vaclav Kotesovec, Sep 03 2018 *)

f(n) = sumdiv(n, k, numdiv(k)); \\ A007425
a(n) = prod(k=1, n, f(k)); \\ Michel Marcus, Mar 23 2021

a(n) = Product_{i=1..n} A007425(i).

a(n) = Product_{prime p<=n} Product_{k=1..floor(log_p(n))} (1 + 2/k)^floor(n/p^k). - Ridouane Oudra, Mar 23 2021

A308912 Highly composite numbers of the 3rd order: numbers k with d_3(k) > d_3(j) for all j < k, where d_3(k) is the 3rd Piltz function, the number of ways of decomposing k into 3 factors (A007425).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 36, 48, 60, 72, 96, 120, 180, 240, 360, 480, 720, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 10080, 15120, 20160, 25200, 30240, 40320, 45360, 50400, 55440, 60480, 75600, 90720, 100800, 110880, 151200, 166320, 221760

Offset: 1

Amiram Eldar, Jun 30 2019

nonn

Pillai defined highly composite numbers of the t-th order and numbers k with d_t(k) > d_t(j) for all j < k, where d_t(k) is the t-th Piltz function, the number of ways of decomposing k into t factors. The highly composite numbers (A002182) are highly composite numbers of the 2nd order.

The corresponding record values are 1, 3, 6, 9, 10, 18, 30, 36, 45, 54, 60, 63, 90, 108, ... (see the link for more terms).

Amiram Eldar, Table of n, a(n) for n = 1..603
Amiram Eldar, Table of n, a(n), d_3(a(n)) for n = 1..603
Jean-Louis Nicolas, On highly composite numbers, in the book Ramanujan Revisited, Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, 1987, Editors G.E. Andrews, R.A. Askey, B.C. Berndt, K.G. Ramanathan, R.A. Rankin.
S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
S. Sivasankaranarayana Pillai, Highly Composite Numbers of the t th Order, J. Indian Math. Soc., Vol. 8 (1944), pp. 61-74.

Cf. A002182, A007425.

f[p_, e_] := (e+1)*(e+2)/2;  a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; am=0; s={}; Do[a1=a[n]; If[a1 > am, am=a1; AppendTo[s, n]], {n, 1, 100000}]; s

A318750 a(n) = Sum_{k=1..n} k * tau_3(k), where tau_3 is A007425.

Original entry on oeis.org

1, 7, 16, 40, 55, 109, 130, 210, 264, 354, 387, 603, 642, 768, 903, 1143, 1194, 1518, 1575, 1935, 2124, 2322, 2391, 3111, 3261, 3495, 3765, 4269, 4356, 5166, 5259, 5931, 6228, 6534, 6849, 8145, 8256, 8598, 8949, 10149, 10272, 11406, 11535, 12327, 13137, 13551

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..100000
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)

Cf. A007425, A034718, A061201, A318413, A318414.

Cf. A001620, A082633.

Accumulate[Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, 100}]]
(* Asymptotics: *) n^2 * (Log[n]^2 + (6*EulerGamma - 1)*Log[n] + 6*EulerGamma^2 - 3*EulerGamma - 6*StieltjesGamma[1] + 1/2) / 4 (* Vaclav Kotesovec, Sep 09 2018 *)

tau_3(k) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(k)[, 2]));
a(n) = sum(k = 1, n,  k * tau_3(k)); \\ Amiram Eldar, Jan 18 2025

a(n) = Sum_{k=1..n} A034718(k).

a(n) ~ n^2 * (log(n)^2 + (6*g-1)*log(n) + 6*g^2 - 3*g - 6*g1 + 1/2) / 4, where g is the Euler-Mascheroni constant A001620 and g1 is the first Stieltjes constant A082633. - Vaclav Kotesovec, Sep 09 2018

A331073 a(n) = Sum_{k = 1..n} [d_3(k)*d_3(n+1-k)], where d_3 = A007425.

Original entry on oeis.org

0, 1, 6, 15, 30, 51, 72, 114, 128, 207, 204, 315, 294, 465, 402, 622, 510, 834, 636, 1023, 834, 1251, 918, 1602, 1074, 1767, 1392, 2117, 1428, 2523, 1620, 2781, 2088, 3003, 2000, 3882, 2238, 3729, 2886, 4350, 2688, 5055, 2928, 5313, 3768, 5343, 3420, 7032, 3678, 6264, 4800, 7347, 4230, 8278, 4584, 8394

Offset: 0

N. J. A. Sloane, Jan 10 2020

nonn

For background references see A330570.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A007425.

See A331074 for another version.

f[p_, e_] := (e+1)*(e+2)/2; s[1] = 1; s[n_] := s[n] = Times @@ f @@@ FactorInteger[n]; a[n_] := Sum[s[k] * s[n+1-k], {k, 1, n}]; Array[a, 100, 0] (* Amiram Eldar, Apr 19 2024 *)

Offset corrected by Amiram Eldar, Apr 19 2024

A331074 a(n) = Sum_{k = 1..ceiling(n/2)} [d_3(k)*d_3(n+1-k)], where d_3 = A007425.

Original entry on oeis.org

0, 1, 3, 12, 15, 30, 36, 75, 64, 108, 102, 198, 147, 237, 201, 361, 255, 435, 318, 552, 417, 630, 459, 963, 537, 888, 696, 1099, 714, 1302, 810, 1503, 1044, 1506, 1000, 2103, 1119, 1869, 1443, 2337, 1344, 2568, 1464, 2697, 1884, 2676, 1710, 3966, 1839, 3150, 2400, 3714, 2115, 4189, 2292, 4359, 2943, 4098

Offset: 0

N. J. A. Sloane, Jan 10 2020

nonn

For background references see A330570.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A007425.

See A331073 for another version.

f[p_, e_] := (e+1)*(e+2)/2; s[1] = 1; s[n_] := s[n] = Times @@ f @@@ FactorInteger[n]; a[n_] := Sum[s[k] * s[n+1-k], {k, 1, Ceiling[n/2]}]; Array[a, 100, 0] (* Amiram Eldar, Apr 19 2024 *)

Offset and name corrected by Amiram Eldar, Apr 19 2024

cp's OEIS Frontend

A061201 Partial sums of A007425: (tau<=)_3(n).

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A077592 Table by antidiagonals of tau_k(n), the k-th Piltz function (see A007425), or n-th term of the sequence resulting from applying the inverse Möbius transform (k-1) times to the all-ones sequence.

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A180365 Partial sums up to 10^n of A007425.

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A308042 Decimal expansion of the asymptotic mean of d_3(k)/ud_3(k), where d_3(k) is the number of ordered factorizations of k as product of 3 divisors (A007425) and ud_3(k) = 3^omega(k) is the unitary analog of d_3 (A074816).

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A331080 a(n) = Sum_{i=1..n} d_3(i)*d_3(i+1), where d_3(n) = A007425(n).

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A175596 Partial products of A007425.

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A308912 Highly composite numbers of the 3rd order: numbers k with d_3(k) > d_3(j) for all j < k, where d_3(k) is the 3rd Piltz function, the number of ways of decomposing k into 3 factors (A007425).

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