A061201 -id:A061201 - cp's OEIS frontend

A211795 Number of (w,x,y,z) with all terms in {1,...,n} and wx < 2y*z.

0, 1, 11, 58, 177, 437, 894, 1659, 2813, 4502, 6836, 10008, 14121, 19449, 26117, 34372, 44422, 56597, 71044, 88160, 108115, 131328, 158074, 188773, 223604, 263172, 307719, 357715, 413493, 475690, 544480, 620632, 704381, 796413

Offset: 0

Clark Kimberling, Apr 27 2012

nonn

Each sequence in the following guide counts 4-tuples
(w,x,y,z) such that the indicated relation holds and the four numbers w,x,y,z are in {1,...,n}.  The notation "m div" means that m divides every term of the sequence.
A211058 ... wx <= yz
A211787 ... wx <= 2yz
A211795 ... wx < 2yz
A211797 ... wx > 2yz
A211809 ... wx >= 2yz
A211812 ... wx <= 3yz
A211917 ... wx < 3yz
A211918 ... wx > 3yz
A211919 ... wx >= 3yz
A211920 ... 2wx < 3yz
A211921 ... 2wx <= 3yz
A211922 ... 2wx > 3yz
A211923 ... 2wx >= 3yz
A212019 ... wx = 2yz ..... 2 div
A212020 ... wx = 3yz ..... 2 div
A212021 ... 2wx = 3yz .... 2 div
A212047 ... wx = 4yz
A212048 ... 3wx = 4yz .... 2 div
A212049 ... wx = 5yz ..... 2 div
A212050 ... 2wx = 5yz .... 2 div
A212051 ... 3wx = 5yz .... 2 div
A212052 ... 4wx = 5yz .... 2 div
A209978 ... wx = yz + 1 .. 2 div
A212053 ... wx <= yz + 1
A212054 ... wx > yz + 1
A212055 ... wx <= yz + 2
A212056 ... wx > yz + 2
A197168 ... wx = yz + 2 .. 2 div
A061201 ... w = xyz
A212057 ... w < xyz
A212058 ... w >= xyz
A212059 ... w = xyz - 1
A212060 ... w = xyz - 2
A212061 ... wx = (yz)^2
A212062 ... w^2 = xyz
A212063 ... w^2 < xyz
A212064 ... w^2 >= xyz
A212065 ... w^2 <= xyz
A212066 ... w^2 > xyz
A212067 ... w^3 = xyz
A002623 ... w = 2x + y + z
A006918 ... w = 2x + 2y + z
A000601 ... w = x + 2y + 3z (except for initial 0's)
A212068 ... 2w = x + y + z
A212069 ... 3w = x + y + z (w = average{x,y,z})
A212088 ... 3w < x + y + z
A212089 ... 3w >= x + y + z
A212090 ... w < x + y + z
A000332 ... w >= x + y + z
A212145 ... w < 2x + y + z
A001752 ... w >= 2x + y + z
A001400 ... w = 2x +3y + 4z
A005900 ... w = -x + y + z
A192023 ... w = -x + y + z + 2
A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div
A212087 ... w^2 + x^2 = y^2 + z^2
A212092 ... w^2 < x^2 + y^2 + z^2
A212093 ... w^2 <= x^2 + y^2 + z^2
A212094 ... w^2 > x^2 + y^2 + z^2
A212095 ... w^2 >= x^2 + y^2 + z^2
A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div
A212097 ... w^3 < x^3 + y^3 + z^3
A212098 ... w^3 <= x^3 + y^3 + z^3
A212099 ... w^3 > x^3 + y^3 + z^3
A212100 ... w^3 >= x^3 + y^3 + z^3
A212101 ... wx^2 = yz^2
A212102 ... 1/w = 1/x + 1/y + 1/z
A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z}
A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m.
A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m.
A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m.
A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m.
A212133 ... median(w,x,y,z) = mean(w,x,y,z)
A212134 ... median(w,x,y,z) <= mean(w,x,y,z)
A212135 ... median(w,x,y,z) > mean(w,x,y,z)
A212241 ... wx + yz > n
A212243 ... 2wx + yz = n
A212244 ... w = xyz - n
A212245 ... w = xyz - 2n
A212246 ... 2w = x + y + z - n
A212247 ... 3w = x + y + z + n
A212249 ... 3w < x + y + z + n
A212250 ... 3w >= x + y + z + n
A212251 ... 3w = x + y + z + n + 1
A212252 ... 3w = x + y + z + n + 2
A212254 ... w = x + 2y + 3z - n
A212255 ... w^2 = mean(x^2, y^2, z^2)
A212256 ... 4/w = 1/x + 1/y +1/z + 1/n
In the list above, if the relation in the second column is of the form "w rel ax + by + cz" then the sequence is linearly recurrent.  In the list below, the same is true for expressions involving more than one relation.
A000332 ... w < x <= y < z .... C(n,4)
A000914 ... w < x <= y < z .... Stirling 1st kind
A000914 ... w < x <= y >= z ... Stirling 1st kind
A050534 ... w < x < y >= z .... tritriangular
A001296 ... w <= x <= y >= z .. 4-dim pyramidal
A006322 ... x < x > y >= z
A002418 ... w < x >= y < z
A050534 ... w < x >=y >= z
A212415 ... w < x >= y <= z
A001296 ... w < x >= y <= z
A212246 ... w <= x > y <= z
A006322 ... w <= x >= y <= z
A212501 ... w > x < y >= z
A212503 ... w < 2x and y < 2z ..... A (note below)
A212504 ... w < 2x and y > 2z ..... A
A212505 ... w < 2x and y >= 2z .... A
A212506 ... w <= 2x and y <= 2z ... A
A212507 ... w < 2x and y <= 2z .... B
A212508 ... w < 2x and y < 3z ..... C
A212509 ... w < 2x and y <= 3z .... C
A212510 ... w < 2x and y > 3z ..... C
A212511 ... w < 2x and y >= 3z .... C
A212512 ... w <= 2x and y < 3z .... C
A212513 ... w <= 2x and y <= 3z ... C
A212514 ... w <= 2x and y > 3z .... C
A212515 ... w <= 2x and y >= 3z ... C
A212516 ... w > 2x and y < 3z ..... C
A212517 ... w > 2x and y <= 3z .... C
A212518 ... w > 2x and y > 3z ..... C
A212519 ... w > 2x and y >= 3z .... C
A212520 ... w >= 2x and y < 3z .... C
A212521 ... w >= 2x and y <= 3z ... C
A212522 ... w >= 2x and y > 3z .... C
A212523 ... w + x < y + z
A212560 ... w + x <= y + z
A212561 ... w + x = 2y + 2z
A212562 ... w + x < 2y + 2z ....... B
A212563 ... w + x <= 2y + 2z ...... B
A212564 ... w + x > 2y + 2z ....... B
A212565 ... w + x >= 2y + 2z ...... B
A212566 ... w + x = 3y + 3z
A212567 ... 2w + 2x = 3y + 3z
A212570 ... |w - x| = |x - y| + |y - z|
A212571 ... |w - x| < |x - y| + |y - z| ... B ... 4 div
A212572 ... |w - x| <= |x - y| + |y - z| .. B
A212573 ... |w - x| > |x - y| + |y - z| ... B ... 2 div
A212574 ... |w - x| >= |x - y| + |y - z| .. B
A212575 ... 2|w - x| = |x - y| + |y - z|
A212576 ... |w - x| = 2|x - y| + 2|y - z|
A212577 ... |w - x| = 2|x - y| - |y - z|
A212578 ... 2|w - x| = |x - y| - |y - z|
A212579 ... min{|w-x|,|w-y|} = min{|x-y|,|x-z|}
A212692 ... w = |x - y| + |y - z| ............... 2 div
A212568 ... w < |x - y| + |y - z| ............... 2 div
A212573 ... w <= |x - y| + |y - z| .............. 2 div
A212574 ... w > |x - y| + |y - z|
A212575 ... w >= |x - y| + |y - z|
A212676 ... w + x = |x - y| + |y - z| ......... H
A212677 ... w + y = |x - y| + |y - z|
A212678 ... w + x + y = |x - y| + |y - z|
A006918 ... w + x + y + z = |x - y| + |y - z| . H
A212679 ... |x - y| = |y - z| ................. H
A212680 ... |x - y| = |y - z| + 1 ..............H 2 div
A212681 ... |x - y| < |y - z| ................... 2 div
A212682 ... |x - y| >= |y - z|
A212683 ... |x - y| = w + |y - z| ............... 2 div
A212684 ... |x - y| = n - w + |y - z|
A212685 ... |w - x| = w + |y - z|
A186707 ... |w - x| < w + |y - z| ... (Note D)
A212714 ... |w - x| >= w + |y - z| .......... H . 2 div
A212686 ... 2*|w - x| = n + |y - z| ............. 4 div
A212687 ... 2*|w - x| < n + |y - z| ......... B
A212688 ... 2*|w - x| < n + |y - z| ......... B . 2 div
A212689 ... 2*|w - x| > n + |y - z| ......... B . 2 div
A212690 ... 2*|w - x| <= n + |y - z| ........ B
A212691 ... w + |x - y| = |x - z| + |y - z| . E . 2 div
...
In the above lists, all the terms of (w,x,y,z) are in {1,...,n}, but in the next lists they are all in {0,...,n}, and sequences are all linearly recurrent.
  R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}.
A212740 ... max{w,x,y,z} < 2*min{w,x,y,z} .... A
A212741 ... max{w,x,y,z} >= 2*min{w,x,y,z} ... A
A212742 ... max{w,x,y,z} <= 2*min{w,x,y,z} ... A
A212743 ... max{w,x,y,z} > 2*min{w,x,y,z} .... A . 2 div
A212744 ... w=range (=max-min) ............... E
A212745 ... w=max{w,x,y,z} - 2*min{w,x,y,z}
A212746 ... R is in {w,x,y,z} ................ E
A212569 ... R is not in {w,x,y,z} ............ E
A212749 ... w=R or xA212750 ... w=R or x=R or yA212751 ... w=R or x=R or yA212752 ... wR ......... A
A212753 ... wR or z>R ......... D
A212754 ... wR or y>R or z>R ......... D
A002415 ... w = x + R ........................ D
A212755 ... |w - x| = R ...................... D
A212756 ... 2w = x + R
A212757 ... 2w = R
A212758 ... w = floor(R/2)
A002413 ... w = floor((x+y+z/2))
A212759 ... w, x, y are even
A212760 ... w is even and x = y + z .......... E
A212761 ... w is odd and x and y are even .... F . 2 div
A212762 ... w and x are odd y is even ........ F . 2 div
A212763 ... w, x, y are odd .................. F
A212764 ... w, x, y are even and z is odd .... F
A030179 ... w and x are even and y and z odd
A212765 ... w is even and x,y,z are odd ...... F
A212766 ... w is even and x is odd ........... A . 2 div
A212767 ... w and x are even and w+x=y+z ..... E
A212889 ... R is even ........................ A
A212890 ... R is odd ......................... A . 2 div
A212742 ... w-x, x-y, y-z are all even ....... A
A212892 ... w-x, x-y, y-z are all odd ........ A
A212893 ... w-x, x-y, y-z have same parity ... A
A005915 ... min{|w-x|, |x-y|, |y-z|} = 0
A212894 ... min{|w-x|, |x-y|, |y-z|} = 1
A212895 ... min{|w-x|, |x-y|, |y-z|} = 2
A179824 ... min{|w-x|, |x-y|, |y-z|} > 0
A212896 ... min{|w-x|, |x-y|, |y-z|} <= 1
A212897 ... min{|w-x|, |x-y|, |y-z|} > 1
A212898 ... min{|w-x|, |x-y|, |y-z|} <= 2
A212899 ... min{|w-x|, |x-y|, |y-z|} > 2
A212901 ... |w-x| = |x-y| = |y-z|
A212900 ... |w-x|, |x-y|, |y-z| are distinct . G
A212902 ... |w-x| < |x-y| < |y-z| ............ G
A212903 ... |w-x| <= |x-y| <= |y-z| .......... G
A212904 ... |w-x| + |x-y| + |y-z| = n ........ H
A212905 ... |w-x| + |x-y| + |y-z| = 2n ....... H
...
Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1.
B: 3,-1,-5,5,1,-3,1
C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1
D: 4,-5,0,5,-4,1
E: 1,3,-3,-3,3,1,-1
F: 1,4,-4,-6,6,4,-4,-1,1
G: 2,1,-3,-1,1,3,-1,-2,1
H: 2,1,-4,1,2,-1

		Examples
					a(2)=11 counts these (w,x,y,z): (1,1,1,1), (1,1,1,2), (1,1,2,1), (2,1,2,1), (2,1,1,2), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,2,2), (2,1,2,2), (2,2,2,2).
		
	
References
			A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

		
Links
			Bo Gyu Jeong, Table of n, a(n) for n = 0..200

		
CrossrefsCf. A000583 (n^4), A210000, A211809, A212959.
Programs
			
				Mathematica
				t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w*x < 2 y*z, s = s + 1], {w, 1, #},
      {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211795 *)
(* Peter J. C. Moses, Apr 13 2012 *)
				

		
Formulaa(n) = n^4 - A211809(n).

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

Original entry on oeis.org

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

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

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

A061204 (tau<=)_6(n).

Original entry on oeis.org

1, 7, 13, 34, 40, 76, 82, 138, 159, 195, 201, 327, 333, 369, 405, 531, 537, 663, 669, 795, 831, 867, 873, 1209, 1230, 1266, 1322, 1448, 1454, 1670, 1676, 1928, 1964, 2000, 2036, 2477, 2483, 2519, 2555, 2891, 2897, 3113, 3119, 3245, 3371, 3407, 3413

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.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#]*DivisorSigma[0, #] &], {n, 1, nmax}]; tau5 = Table[Sum[tau4[[d]], {d, Divisors[n]}], {n, nmax}]; Accumulate[Table[Sum[tau5[[d]], {d, Divisors[n]}], {n, nmax}]] (* Vaclav Kotesovec, Sep 10 2018 *)

(tau<=)k(n)=Sum{i=1..n} tau_k(i). a(n)=partial sums of A034695.
a(n) = Sum_{k=1..n} tau_{5}(k) * floor(n/k), where tau_{5} is A061200. - Enrique Pérez Herrero, Jan 23 2013
a(n) ~ n*(log(n)^5/120 + (g/4 - 1/24)*log(n)^4 + (5*g^2/2 - g - g1 + 1/6)*log(n)^3 + (10*g^3 - 15*g^2/2 + (3 - 15*g1)*g + 3*g1 + 3*g2/2 - 1/2)*log(n)^2 + (15*g^4 - 20*g^3 + (15 - 60*g1)*g^2 + (30*g1 + 15*g2 - 6)*g + 15*g1^2 - 6*g1 - 3*g2 - g3 + 1)*log(n) + 6*g^5 - 15*g^4 + (20 - 60*g1)*g^3 + (60*g1 + 30*g2 - 15)*g^2 + (60*g1^2 - 30*g1 - 15*g2 - 5*g3 + 6)*g - 15*g1^2 + g1*(6 - 15*g2) + 3*g2 + g3 + g4/4 - 1), where g is the Euler-Mascheroni constant A001620 and g1, g2, g3, g4 are the Stieltjes constants, see A082633, A086279, A086280 and A086281. - Vaclav Kotesovec, Sep 10 2018

A356045 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} sigma_k(j) * floor(n/j).

Original entry on oeis.org

1, 1, 4, 1, 5, 7, 1, 7, 10, 13, 1, 11, 18, 21, 16, 1, 19, 40, 45, 28, 25, 1, 35, 102, 123, 72, 48, 28, 1, 67, 280, 393, 250, 138, 57, 38, 1, 131, 798, 1371, 1020, 540, 189, 83, 44, 1, 259, 2320, 5025, 4498, 2514, 885, 301, 101, 53, 1, 515, 6822, 18963, 20652, 12828, 4917, 1553, 403, 129, 56

Offset: 1

Seiichi Manyama, Jul 24 2022

			Square array begins:
   1,  1,   1,   1,    1,     1, ...
   4,  5,   7,  11,   19,    35, ...
   7, 10,  18,  40,  102,   280, ...
  13, 21,  45, 123,  393,  1371, ...
  16, 28,  72, 250, 1020,  4498, ...
  25, 48, 138, 540, 2514, 12828, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=0..3 give A061201, A280077, A356042, A356043.
T(n,n) gives A356046.
Cf. A322103.

T(n, k) = sum(j=1, n, sigma(j, k)*(n\j));

T(n, k) = sum(j=1, n, sumdiv(j, d, d^k*numdiv(j/d)));

T(n, k) = sum(j=1, n, sumdiv(j, d, sigma(d, k)));

G.f. of column k: (1/(1-x)) * Sum_{j>=1} sigma_k(j) * x^j/(1 - x^j).
T(n,k) = Sum_{j=1..n} Sum_{d|j} d^k * tau(j/d).
T(n,k) = Sum_{j=1..n} Sum_{d|j} sigma_k(d).

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

A212057 Number of (w,x,y,z) with all terms in {1,...,n} and w

Original entry on oeis.org

0, 0, 11, 69, 231, 584, 1230, 2307, 3964, 6385, 9771, 14356, 20377, 28125, 37894, 50008, 64809, 82681, 104005, 129216, 158743, 193063, 232668, 278080, 329812, 388452, 454585, 528822, 611791, 704167, 806610, 919852, 1044607, 1181643

Offset: 0

Clark Kimberling, Apr 30 2012

nonn

a(n)+A212058(n)=n^4. For a guide to related sequences, see A211795.

Cf. A211795, A061201.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212057 *)
(* Peter J. C. Moses, Apr 13 2012 *)

Showing 1-10 of 22 results. Next

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A355541 Numbers k such that A061201(k) is divisible by k.

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A211795 Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z.

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

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

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A061202 (tau<=)_4(n).

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A061203 (tau<=)_5(n).

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A061204 (tau<=)_6(n).

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A211795 Number of (w,x,y,z) with all terms in {1,...,n} and wx < 2y*z.