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 1-3 of 3 results.

A064950 a(n) = Sum_{i|n, j|n} lcm(i,j).

Original entry on oeis.org

1, 7, 10, 27, 16, 70, 22, 83, 55, 112, 34, 270, 40, 154, 160, 227, 52, 385, 58, 432, 220, 238, 70, 830, 141, 280, 244, 594, 88, 1120, 94, 579, 340, 364, 352, 1485, 112, 406, 400, 1328, 124, 1540, 130, 918, 880, 490, 142, 2270, 267, 987, 520, 1080, 160, 1708
Offset: 1

Views

Author

Vladeta Jovovic, Oct 28 2001

Keywords

Links

Crossrefs

Cf. A060724, A000005, A062369, A062368, A062380.

Programs

  • Mathematica
    a[n_]:= Sum[LCM[i,j], {i, Divisors[n]}, {j, Divisors[n]}];
Array[a,60] (* Jean-François Alcover, Jun 03 2019 *)
f[p_, e_] := (p^(e+2) - 3*p^(e+1) + p + 1 + 2*p^(e+2)*e - 2*p^(e+1)*e)/(p-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60] (* Amiram Eldar, Aug 28 2023 *)
  • PARI
    for (n=1, 1000, d=divisors(n); a=sum(i=1, length(d), numdiv(d[i]^2)*d[i]); write("b064950.txt", n, " ", a)) \\ Harry J. Smith, Oct 01 2009
  • Sage
    def A064950(n) :
    tau = sloane.A000005; D = divisors(n)
    return reduce(lambda x,y: x+y, [d*tau(d^2) for d in D])
[A064950(n) for n in (1..54)] # Peter Luschny, Sep 10 2012

Formula

a(n) = Sum_{d|n} d*tau(d^2).
Multiplicative with a(p^e) = (p^(e+2) - 3*p^(e+1) + p + 1 + 2*p^(e+2)*e - 2*p^(e+1)*e)/(p-1)^2.

A062367 Multiplicative with a(p^e) = (e+1)*(e+2)*(2*e+3)/6.

Original entry on oeis.org

1, 5, 5, 14, 5, 25, 5, 30, 14, 25, 5, 70, 5, 25, 25, 55, 5, 70, 5, 70, 25, 25, 5, 150, 14, 25, 30, 70, 5, 125, 5, 91, 25, 25, 25, 196, 5, 25, 25, 150, 5, 125, 5, 70, 70, 25, 5, 275, 14, 70, 25, 70, 5, 150, 25, 150, 25, 25, 5, 350, 5, 25, 70, 140, 25, 125, 5, 70, 25, 125, 5
Offset: 1

Views

Author

Vladeta Jovovic, Jul 07 2001

Keywords

Links

Crossrefs

Cf. A000005, A000012, A000330, A029939, A035116, A048691, A060648, A062368.

Programs

  • Maple
    A062367 := proc(n)
    add(numtheory[tau](d)^2,d=numtheory[divisors](n)) ;
end proc:
seq(A062367(n),n=1..40) ; # R. J. Mathar, May 15 2025
  • Mathematica
    {1}~Join~Array[Times @@ Map[((# + 1) (# + 2) (2 # + 3))/6 &, FactorInteger[#][[All, -1]] ] &, 70, 2] (* or *)
Array[DivisorSum[#, DivisorSigma[0, #]^2 &] &, 71] (* Michael De Vlieger, Mar 05 2021 *)
  • PARI
    a(n) = sumdiv(n, d, numdiv(d)^2) \\ Michel Marcus, Jun 17 2013

Formula

a(n) = Sum_{i|n, j|n} tau(gcd(i, j)) = Sum_{d|n} tau(d)^2.
a(n) = Sum_{i|n, j|n} tau(i)*tau(j)/tau(lcm(i, j)), where tau(n) = number of divisors of n, cf. A000005.
Dirichlet convolution of A035116 and A000012 (i.e., inverse Mobius transform of A035116). Dirichlet g.f.: zeta^5(s)/zeta(2s). - R. J. Mathar, Feb 03 2011
G.f.: Sum_{n>=1} A000005(n)^2*x^n/(1-x^n). - Mircea Merca, Feb 26 2014
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(tau(k)^2/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
Dirichlet convolution of A007426 and A008966. Dirichlet convolution of A007425 and A034444. - R. J. Mathar, Jun 05 2020
Let b(n), n > 0, be Dirichlet inverse of a(n). Then b(n) is multiplicative with b(p^e) = (-1)^e*(Sum_{i=0..e} binomial(4,i)) for prime p and e >= 0, where binomial(n,k)=0 if n < k; abs(b(n)) is multiplicative and has the Dirichlet g.f.: (zeta(s))^5/(zeta(2*s))^4. - Werner Schulte, Feb 07 2021
a(n) = Sum_{d divides n} tau(d^2)*tau(n/d), Dirichlet convolution of A048691 and A000005. - Peter Bala, Jan 26 2024

A062369 Dirichlet convolution of n and tau^2(n).

Original entry on oeis.org

1, 6, 7, 21, 9, 42, 11, 58, 30, 54, 15, 147, 17, 66, 63, 141, 21, 180, 23, 189, 77, 90, 27, 406, 54, 102, 106, 231, 33, 378, 35, 318, 105, 126, 99, 630, 41, 138, 119, 522, 45, 462, 47, 315, 270, 162, 51, 987, 86, 324, 147, 357, 57, 636, 135, 638, 161, 198, 63, 1323
Offset: 1

Views

Author

Vladeta Jovovic, Jul 07 2001

Keywords

Comments

Dirichlet convolution of A000027 and A035116.
Inverse Mobius transform of A060724. - R. J. Mathar, Oct 15 2011

Links

Crossrefs

Cf. A000203, A060648.
Cf. A000005, A060724, A064950, A062369, A062368, A062380.

Programs

  • Magma
    [&+[d*#Divisors(Floor(n/d))^2:d in Divisors(n)]:n in [1..60]]; // Marius A. Burtea, Aug 25 2019
  • Mathematica
    a[n_] := Sum[ DivisorSigma[1, i]*DivisorSigma[1, j] / DivisorSigma[1, LCM[i, j]], {i, Divisors[n]}, {j, Divisors[n]}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 26 2013 *)
  • PARI
    a(n) = sumdiv(n, d, d*numdiv(n/d)^2); \\ Michel Marcus, Nov 03 2018

Formula

a(n) = Sum_{i|n, j|n} sigma(i)*sigma(j)/sigma(lcm(i,j)), where sigma(n) = sum of divisors of n.
a(n) = Sum_{i|d, j|d} sigma(gcd(i, j));
a(n) = Sum_{d|n} d*tau(n/d)^2, where tau(n) = number of divisors of n.
Multiplicative with a(p^e) = (1-p^(3+e)-p^(2+e)+e^2+4*p^2+p^2*e^2+2*e-3*p+4*p^2*e-6*e*p-2*e^2*p)/(1-p)^3.
Dirichlet g.f.: (zeta(s))^4*zeta(s-1)/zeta(2*s). - R. J. Mathar, Feb 09 2011
G.f.: Sum_{k>=1} tau(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
Sum_{k=1..n} a(k) ~ 5 * Pi^4 * n^2 / 144. - Vaclav Kotesovec, Jan 28 2019
a(n) = Sum_{d|n} tau(d^2)*sigma(n/d), where tau(n) = number of divisors of n, and sigma(n) = sum of divisors of n. - Ridouane Oudra, Aug 25 2019
Showing 1-3 of 3 results.

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