cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

N. J. A. Sloane, May 24 1994

Keywords

Comments

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

Examples

			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

References

  • 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).

Links

Crossrefs

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.

Programs

  • Haskell
    a007425 = sum . map a000005 . a027750_row
-- Reinhard Zumkeller, Feb 16 2012
  • Maple
    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
  • Mathematica
    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 *)
  • 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
  • PARI
    a(n)=sumdivmult(n,k,numdiv(k)) \\ Charles R Greathouse IV, Aug 30 2013
  • PARI
    for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^3)[n]), ", ")) \\ Vaclav Kotesovec, May 06 2025
  • Python
    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

Formula

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

A085780 Numbers that are a product of 2 triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 9, 10, 15, 18, 21, 28, 30, 36, 45, 55, 60, 63, 66, 78, 84, 90, 91, 100, 105, 108, 120, 126, 135, 136, 150, 153, 165, 168, 171, 190, 198, 210, 216, 225, 231, 234, 253, 270, 273, 276, 280, 300, 315, 325, 330, 351, 360, 378, 396, 406, 408, 420, 435, 441
Offset: 1

Views

Author

Jon Perry, Jul 23 2003

Keywords

Comments

Is there a fast algorithm for detecting these numbers? - Charles R Greathouse IV, Jan 26 2013
The number of rectangles with positive width 1<=w<=i and positive height 1<=h<=j contained in an i*j rectangle is t(i)*t(j), where t(k)=A000217(k), see A096948. - Dimitri Boscainos, Aug 27 2015

Examples

			18 = 3*6 = t(2)*t(3) is a product of two triangular numbers and therefore in the sequence.

Links

Crossrefs

Cf. A000217, A085782, A068143, A000537 (subsequence), A006011 (subsequence), A033487 (subsequence), A188630 (subsequence).
Cf. A072389 (this times 4).

Programs

  • Maple
    isA085780 := proc(n)
     local d;
     for d in numtheory[divisors](n) do
        if d^2 > n then
            return false;
        end if;
        if isA000217(d) then
            if isA000217(n/d) then
                return true;
            end if;
        end if;
    end do:
    return false;
end proc:
for n from 1 to 1000 do
    if isA085780(n) then
        printf("%d,",n) ;
    end if ;
end do: # R. J. Mathar, Nov 29 2015
  • Mathematica
    t1 = Table[n (n+1)/2, {n, 0, 100}];Select[Union[Flatten[Outer[Times, t1, t1]]], # <= t1[[-1]] &] (* T. D. Noe, Jun 04 2012 *)
  • PARI
    A003056(n)=(sqrtint(8*n+1)-1)\2
list(lim)=my(v=List([0]),t); for(a=1, A003056(lim\1), t=a*(a+1)/2; for(b=a, A003056(lim\t), listput(v,t*b*(b+1)/2))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jan 26 2013
  • Python
    from itertools import count, islice
from sympy import divisors, integer_nthroot
def A085780_gen(startvalue=0): # generator of terms
    if startvalue <= 0:
        yield 0
    for n in count(max(startvalue,1)):
        for d in divisors(m:=n<<2):
            if d**2 > m:
                break
            if integer_nthroot((d<<2)+1,2)[1] and integer_nthroot((m//d<<2)+1,2)[1]:
                yield n
                break
A085780_list = list(islice(A085780_gen(),10)) # Chai Wah Wu, Aug 28 2022

Formula

Conjecture: There are about sqrt(x)*log(x) terms up to x. - Charles R Greathouse IV, Jul 11 2024

Extensions

More terms from Max Alekseyev and Jon E. Schoenfield, Sep 04 2009

A308264 Number of ordered factorizations of n into triangular numbers > 1.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, May 17 2019

Keywords

Links

Crossrefs

Cf. A000217, A010054, A085782 (for n > 1, positions of nonzero terms).
Not the absolute values of A365800.

Programs

  • Mathematica
    terms = 100; A[] = 0; Do[A[x] = x + Sum[A[x^(k (k + 1)/2)], {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := If[n == 1, n, Sum[If[(d < n && IntegerQ[Sqrt[8 n/d + 1]]), a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 100}]
  • PARI
    A010054(n) = issquare(1 + 8*n);
A308264(n) = if(1==n,n,sumdiv(n,d,if(dA010054(n/d)*A308264(d),0))); \\ Antti Karttunen, Oct 05 2023

Formula

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} A(x^(k*(k+1)/2)).
a(1) = 1; a(n) = Sum_{d|n, dA010054(n/d)*a(d).

Extensions

Data section extended to 105 terms by Antti Karttunen, Oct 05 2023

A334115 Numbers that can be written as a product of tetrahedral numbers.

Original entry on oeis.org

0, 1, 4, 10, 16, 20, 35, 40, 56, 64, 80, 84, 100, 120, 140, 160, 165, 200, 220, 224, 256, 286, 320, 336, 350, 364, 400, 455, 480, 560, 640, 660, 680, 700, 800, 816, 840, 880, 896, 969, 1000, 1024, 1120, 1140, 1144, 1200, 1225, 1280, 1330, 1344, 1400, 1456, 1540, 1600
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 14 2020

Keywords

Comments

For n > 1, numbers that appear at least once in A007426.

Links

Crossrefs

Cf. A000292, A007426, A085782, A265010.

Extensions

More terms from David A. Corneth, Mar 22 2021

A334129 Numbers that can be written as a product of one or more consecutive triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 18, 21, 28, 36, 45, 55, 60, 66, 78, 91, 105, 120, 136, 150, 153, 171, 180, 190, 210, 231, 253, 276, 300, 315, 325, 351, 378, 406, 435, 465, 496, 528, 561, 588, 595, 630, 666, 703, 741, 780, 820, 861, 900, 903, 946, 990, 1008, 1035
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 14 2020

Keywords

Links

Crossrefs

Cf. A000217, A006472, A085780, A085782, A334130.

Programs

  • Mathematica
    lmt = 1050; t = PolygonalNumber[3, #] & /@ Range[0, Sqrt[ 2lmt]]; f[n_] := Select[ Times @@@ Partition[t, n +1, 1], # < lmt &]; lst = {}; k = 0; While[f@k != {}, lst = Join[lst, f@k]; k++]; Union@lst (* Robert G. Wilson v, Apr 16 2020 *)
  • PARI
    list(lim)=if(lim<1, return(if(lim<0,[],[0]))); my(v=List([0,1]),t=1,m=2); lim\=1; while(t<=lim, listput(v,t); t=m*m++/2); for(e=1,m, for(i=3,m-e, t=factorback(Vec(v[i..i+e])); if(t>lim, break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

A334130 Numbers that can be written as a product of distinct triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 18, 21, 28, 30, 36, 45, 55, 60, 63, 66, 78, 84, 90, 91, 105, 108, 120, 126, 135, 136, 150, 153, 165, 168, 171, 180, 190, 198, 210, 216, 231, 234, 253, 270, 273, 276, 280, 300, 315, 325, 330, 351, 360, 378, 396, 406, 408, 420, 435, 450
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 14 2020

Keywords

Links

Crossrefs

Cf. A000217, A006472, A054731, A085780, A085782, A334129.

Programs

  • Maple
    N:= 1000: # for all terms <= N
S:= {0,1}:
for i from 2 do
  t:= i*(i+1)/2;
  if t > N then break fi;
  S:= S union select(`<=`,map(`*`,S,t),N)
od:
sort(convert(S,list)); # Robert Israel, Apr 21 2020
Showing 1-6 of 6 results.

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