A061204 -id:A061204 - 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

A061200 tau_5(n) = number of ordered 5-factorizations of n.

Original entry on oeis.org

1, 5, 5, 15, 5, 25, 5, 35, 15, 25, 5, 75, 5, 25, 25, 70, 5, 75, 5, 75, 25, 25, 5, 175, 15, 25, 35, 75, 5, 125, 5, 126, 25, 25, 25, 225, 5, 25, 25, 175, 5, 125, 5, 75, 75, 25, 5, 350, 15, 75, 25, 75, 5, 175, 25, 175, 25, 25, 5, 375, 5, 25, 75, 210, 25, 125, 5, 75, 25, 125, 5

Offset: 1

Vladeta Jovovic, Apr 21 2001

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)

Cf. tau_2(n): A000005, tau_3(n): A007425, tau_4(n): A007426, tau_6(n): A034695, (unordered) 2-factorization of n: A038548, (unordered) 3-factorization of n: A034836, A001055, A006218, A061201, A061202, A061203 (partial sums), A061204.

Column k=5 of A077592.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 5], {n, 77}] (* Robert G. Wilson v *)
tau[1, k_] := 1; tau[n_, k_] := Times @@ (Binomial[Last[#]+k-1, k-1]& /@ FactorInteger[n]); Table[tau[n, 5], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)

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

a(n)=sumdivmult(n,k,sumdivmult(k,x,sumdivmult(x,y,numdiv(y)))) \\ Charles R Greathouse IV, Sep 09 2014

a(n, f=factor(n))=f=f[, 2]; prod(i=1, #f, binomial(f[i]+4, 4)) \\ Charles R Greathouse IV, Oct 28 2017

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^5)[n]), ", ")) \\ Vaclav Kotesovec, May 06 2025

from math import prod, comb
from sympy import factorint
def A061200(n): return prod(comb(4+e,4) for e in factorint(n).values()) # Chai Wah Wu, Dec 22 2024

tau_k(n) = |{(x_1,x_2,...,x_k): x_1*x_2*...*x_k=n}|, number of ordered k-factorizations of n.
tau_k(p^m) = (-1)^(k-1)*binomial(-m-1,k-1), p prime.
limit(tau_k(n)/n^epsilon, n=infinity) = 0, for any epsilon>0.
tau_k(n) = Sum_{d|n} tau_(k-1)(d), tau_1(n)=1.
Dirichlet g.f.: (zeta(s))^k.
For explicit formula, see A007425.
G.f.: Sum_{k>=1} tau_4(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

A034695 Tau_6 (the 6th Piltz divisor function), the number of ordered 6-factorizations of n; Dirichlet convolution of number-of-divisors function (A000005) with A007426.

Original entry on oeis.org

1, 6, 6, 21, 6, 36, 6, 56, 21, 36, 6, 126, 6, 36, 36, 126, 6, 126, 6, 126, 36, 36, 6, 336, 21, 36, 56, 126, 6, 216, 6, 252, 36, 36, 36, 441, 6, 36, 36, 336, 6, 216, 6, 126, 126, 36, 6, 756, 21, 126, 36, 126, 6, 336, 36, 336, 36, 36, 6, 756, 6, 36, 126, 462, 36, 216, 6, 126

Offset: 1

N. J. A. Sloane

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pages 29 and 38
Leveque, William J., Fundamentals of Number Theory. New York:Dover Publications, 1996, ISBN 9780486689067, p .167-Exercise 5.b.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)
E. Pérez Herrero, Piltz Divisor functions (1), Psychedelic Geometry Blogspot, Dec 21 2009.
E. Pérez Herrero, Piltz Divisor functions (2), Psychedelic Geometry Blogspot, Dec 24 2009.

Cf. A000005 (tau_2), A007425 (tau_3), A007426 (tau_4), A061200 (tau_5).
Cf. A061204.
Column k=6 of A077592.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[ #, k - 1] & /@ Divisors[n]); Table[ tau[n, 6], {n, 68}] (* 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, 6], {n, 1, 100}] (* Amiram Eldar, Sep 13 2020 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,1] = binomial(f[i,2] + 5, f[i,2]); f[i,2]=1); factorback(f); \\ Michel Marcus, Jun 09 2014

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^6)[n]), ", ")) \\ Vaclav Kotesovec, May 06 2025

from math import prod, comb
from sympy import factorint
def A034695(n): return prod(comb(5+e,5) for e in factorint(n).values()) # Chai Wah Wu, Dec 22 2024

Dirichlet g.f.: zeta^6(s).
Multiplicative with a(p^e) = binomial(e+5, e). - Mitch Harris, Jun 27 2005
The Piltz divisor functions hold for tau_j(*)tau_k = tau_{j+k}, where (*) means Dirichlet convolution.
G.f.: Sum_{k>=1} tau_5(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Oct 30 2018

More terms from Robert G. Wilson v, Nov 02 2005

A061202 (tau<=)_4(n).

Original entry on oeis.org

1, 5, 9, 19, 23, 39, 43, 63, 73, 89, 93, 133, 137, 153, 169, 204, 208, 248, 252, 292, 308, 324, 328, 408, 418, 434, 454, 494, 498, 562, 566, 622, 638, 654, 670, 770, 774, 790, 806, 886, 890, 954, 958, 998, 1038, 1054, 1058, 1198, 1208, 1248, 1264, 1304, 1308

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.
Partial sums of A007426.
Equals row sums of triangle A140703. - Gary W. Adamson, May 24 2008

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (1000000 terms)
Eric Weisstein's World of Mathematics, Stieltjes Constants
Wikipedia, Stieltjes constants

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<=)_3(n): A061201, (tau<=)_5(n): A061203, (tau<=)_6(n): A061204.
Equals left column of triangle A140705.
Cf. A140703.

(* Asymptotics: *) n * (Log[n]^3/6 + (2*EulerGamma - 1/2)*Log[n]^2 + (6*EulerGamma^2 - 4*EulerGamma - 4*StieltjesGamma[1] + 1)*Log[n] + 4*EulerGamma^3 - 6*EulerGamma^2 + 4*EulerGamma + 4*StieltjesGamma[1]*(1 - 3*EulerGamma) + 2*StieltjesGamma[2] - 1) (* Vaclav Kotesovec, Sep 09 2018 *)

(tau<=)k(n) = Sum{i=1..n} tau_k(i).
a(n) = Sum_{k = 1..n} tau_{3}(k)*floor (n/k), where tau_{3} is A007425. - Enrique Pérez Herrero, Jan 23 2013
a(n) ~ n * (log(n)^3/6 + (2*g - 1/2)*log(n)^2 + (6*g^2 - 4*g - 4*g1 + 1)*log(n) + 4*g^3 - 6*g^2 + 4*g + 4*g1*(1 - 3*g) + 2*g2 - 1), where g is the Euler-Mascheroni constant A001620, g1 and g2 are the Stieltjes constants, see A082633 and A086279. - Vaclav Kotesovec, Sep 09 2018
a(n) = Sum_{i=1..n} tau(i)*A006218(floor(n/i)). - Ridouane Oudra, Sep 17 2021
a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} floor(n/(i*j*k)). - Ridouane Oudra, Oct 31 2022

A061203 (tau<=)_5(n).

Original entry on oeis.org

1, 6, 11, 26, 31, 56, 61, 96, 111, 136, 141, 216, 221, 246, 271, 341, 346, 421, 426, 501, 526, 551, 556, 731, 746, 771, 806, 881, 886, 1011, 1016, 1142, 1167, 1192, 1217, 1442, 1447, 1472, 1497, 1672, 1677, 1802, 1807, 1882, 1957, 1982, 1987, 2337, 2352

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.
Partial sums of A061200.
Equals row sums of triangle A140705. - Gary W. Adamson, May 24 2008

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 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<=)_3(n): A061201, (tau<=)_4(n): A061202, (tau<=)_6(n): A061204.

Cf. A140705.

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(5, n)+a(n-1)) end:
seq(a(n), n=1..49);  # Alois P. Heinz, Feb 13 2022

nmax = 50;
tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #] &], {n, 1, nmax}];
Accumulate[Table[Sum[tau4[[d]], {d, Divisors[n]}], {n, nmax}]] (* Vaclav Kotesovec, Sep 10 2018 *)

(tau<=)k(n) = Sum{i=1..n} tau_k(i).
a(n) = Sum_{k=1..n} tau_{4}(k) * floor(n/k), where tau_{4} is A007426. - Enrique Pérez Herrero, Jan 23 2013
a(n) ~ n*(log(n)^4/24 + (5*g/6 - 1/6)*log(n)^3 + 10*g1^2 + (5*g^2 - 5*g/2 - 5*g1/2 + 1/2)*log(n)^2 + (10*g^3 - 10*g^2 + (5 - 20*g1)*g + 5*g1 + 5*g2/2 - 1)*log(n) + 5*g^4 - 10*g^3 + (10 - 30*g1)*g^2 + (20*g1 + 10*g2 - 5)*g - 5*g1 - 5*g2/2 - 5*g3/6 + 1), where g is the Euler-Mascheroni constant A001620 and g1, g2, g3 are the Stieltjes constants, see A082633, A086279 and A086280. - Vaclav Kotesovec, Sep 10 2018

A077593 Table by antidiagonals where T(n,k) = Sum_{i=1..n} T(floor(n/i),k-1) starting with T(n,0)=1 if n>0 and T(0,0)=0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 5, 4, 1, 0, 1, 5, 7, 8, 5, 1, 0, 1, 6, 9, 13, 10, 6, 1, 0, 1, 7, 11, 19, 16, 14, 7, 1, 0, 1, 8, 13, 26, 23, 25, 16, 8, 1, 0, 1, 9, 15, 34, 31, 39, 28, 20, 9, 1, 0, 1, 10, 17, 43, 40, 56, 43, 38, 23, 10, 1, 0, 1, 11, 19, 53, 50, 76, 61, 63

Offset: 0

Henry Bottomley, Nov 08 2002

			Rows start:
 0,0,0,0,0,0...;
 1,1,1,1,1,1...;
 1,2,3,4,5,6...;
 1,3,5,7,9,11...;
 1,4,8,13,19,26,...;
 ...

Columns include A057427, A001477, A006218, A061201, A061202, A061203, A061204.
Rows include (with offsets) A000004, A000012, A000027, A005408, A034856, A052905.
Cf. A077593.

T(n, k) = T(n-1, k) + A077592(n, k). Writing m as Sum_{i} p_i^e_i, T(n, k) = Sum_{m=1..n} Product_{i} C(k+e_i-1, e_i).

cp's OEIS Frontend

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

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A061200 tau_5(n) = number of ordered 5-factorizations of n.

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A034695 Tau_6 (the 6th Piltz divisor function), the number of ordered 6-factorizations of n; Dirichlet convolution of number-of-divisors function (A000005) with A007426.

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A061202 (tau<=)_4(n).

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A061203 (tau<=)_5(n).

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A077593 Table by antidiagonals where T(n,k) = Sum_{i=1..n} T(floor(n/i),k-1) starting with T(n,0)=1 if n>0 and T(0,0)=0.

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