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.

A351265 Sum of the squares of the squarefree divisors of n.

Original entry on oeis.org

1, 5, 10, 5, 26, 50, 50, 5, 10, 130, 122, 50, 170, 250, 260, 5, 290, 50, 362, 130, 500, 610, 530, 50, 26, 850, 10, 250, 842, 1300, 962, 5, 1220, 1450, 1300, 50, 1370, 1810, 1700, 130, 1682, 2500, 1850, 610, 260, 2650, 2210, 50, 50, 130, 2900, 850, 2810, 50, 3172, 250, 3620
Offset: 1

Views

Author

Wesley Ivan Hurt, Feb 05 2022

Keywords

Comments

Inverse Möbius transform of n^2 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

Examples

			a(6) = 50; a(6) = Sum_{d|6} d^2 * mu(d)^2 = 1^2*1 + 2^2*1 + 3^2*1 + 6^2*1 = 50.

Links

Crossrefs

Cf. A008683 (mu), A253905, A328639, A322360.
Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), this sequence (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

Programs

  • Mathematica
    a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^2); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
Table[Total[Select[Divisors[n],SquareFreeQ]^2],{n,80}] (* Harvey P. Dale, Dec 26 2024 *)
  • PARI
    a(n) = sumdiv(n, d, if (issquarefree(d), d^2)); \\ Michel Marcus, Feb 06 2022

Formula

a(n) = Sum_{d|n} d^2 * mu(d)^2.
a(n) = abs(A328639(n)).
G.f.: Sum_{k>=1} mu(k)^2 * k^2 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Multiplicative with a(p^e) = 1 + p^2. - Amiram Eldar, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/(3*zeta(2)) = A253905 / 3 = 0.243587... . - Amiram Eldar, Nov 10 2022
Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2s-4). - Michael Shamos, Aug 05 2023

A328640 Dirichlet g.f.: zeta(2*s) / (zeta(s) * zeta(s-3)).

Original entry on oeis.org

1, -9, -28, 9, -126, 252, -344, -9, 28, 1134, -1332, -252, -2198, 3096, 3528, 9, -4914, -252, -6860, -1134, 9632, 11988, -12168, 252, 126, 19782, -28, -3096, -24390, -31752, -29792, -9, 37296, 44226, 43344, 252, -50654, 61740, 61544, 1134, -68922, -86688, -79508, -11988, -3528
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 23 2019

Keywords

Comments

Dirichlet inverse of A065959.

Links

Crossrefs

Cf. A008683, A008836, A026424 (positions of negative terms), A063453, A065959, A323363, A328639.

Programs

  • Mathematica
    a[1] = 1; a[n_] := -Sum[DirichletConvolve[j^3, MoebiusMu[j]^2, j, n/d] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 45}]
Table[DivisorSum[n, LiouvilleLambda[n/#] MoebiusMu[#] #^3 &], {n, 1, 45}]
f[p_, e_] := (-1)^e*(p^3+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 03 2022 *)
  • PARI
    a(n)={sumdiv(n, d, (-1)^bigomega(n/d)*moebius(d)*d^3)} \\ Andrew Howroyd, Oct 25 2019

Formula

a(1) = 1; a(n) = -Sum_{d|n, dA065959(n/d) * a(d).
a(n) = Sum_{d|n} lambda(n/d) * mu(d) * d^3, where lambda = A008836 and mu = A008683.
Multiplicative with a(p^) = (-1)^e*(p^3+1). - Amiram Eldar, Dec 03 2022

A372620 Expansion of Sum_{k>=1} k * prime(k) * x^prime(k) / (1 - x^prime(k)).

Original entry on oeis.org

0, 2, 6, 2, 15, 8, 28, 2, 6, 17, 55, 8, 78, 30, 21, 2, 119, 8, 152, 17, 34, 57, 207, 8, 15, 80, 6, 30, 290, 23, 341, 2, 61, 121, 43, 8, 444, 154, 84, 17, 533, 36, 602, 57, 21, 209, 705, 8, 28, 17, 125, 80, 848, 8, 70, 30, 158, 292, 1003, 23, 1098, 343, 34, 2, 93
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2024

Keywords

Examples

			a(60) = a(2^2 * 3 * 5) = a(prime(1)^2 * prime(2) * prime(3)) = 1 * 2 + 2 * 3 + 3 * 5 = 23.

Links

Crossrefs

Cf. A000720, A001414, A005063, A008472, A056239, A061150, A066328, A328639.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[Sum[k Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Plus @@ (PrimePi[#[[1]]] #[[1]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]

Formula

L.g.f.: -log( Product_{k>=1} (1 - x^prime(k))^k ).
If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j) * p_j), where pi = A000720.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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