A198286 a(n) = Sum_{d|n} (A053143(d) or smallest square divisible by d).

1, 5, 10, 9, 26, 50, 50, 25, 19, 130, 122, 90, 170, 250, 260, 41, 290, 95, 362, 234, 500, 610, 530, 250, 51, 850, 100, 450, 842, 1300, 962, 105, 1220, 1450, 1300, 171, 1370, 1810, 1700, 650, 1682, 2500, 1850, 1098, 494, 2650, 2210, 410, 99, 255, 2900, 1530, 2810

Offset

1,2

Comments

Multiplicative function with a(p^e) = 1+2*(p^(e+2)-p^2)/(p^2-1) if e is even else a(p^e)=(1+p^2)((p^(e+1)-1)/(p^2-1)). Examples: a(9)=a(3^2)=1+2*((81-9)/(9-1))=1+2*9=19; a(8)=a(2^3)=(1+4)((16-1)/(4-1))=5*5=25.

Another definition of a(n): Sum_{d|n} (d*core(d)), where core(d) is the squarefree part of d (A007913), i.e., inverse Mobius transform of A053143.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

Dirichlet g.f.: zeta(s)*zeta(s-2)*zeta(2s-2)/zeta(2s-4). - R. J. Mathar, Mar 12 2012

Sum_{k=1..n} a(k) ~ Pi^2 * Zeta(3) * n^3 / 45. - Vaclav Kotesovec, Feb 02 2019

Example

a(18) = 95 because 18=2*3^2, so a(18) = (1+4)(1+9+9) = 5*19 = 95.
a(20) = 234 because 20=2^2*5, so a(20) = (1+4+4)(1+25) = 9*26 = 234.

Mathematica

ssq[n_] := For[k=1, True, k++, If[ Divisible[s = k^2, n], Return[s]]]; a[n_] := Sum[ ssq[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 53}] (* Jean-François Alcover, Sep 03 2012 *)
f[p_, e_] := If[OddQ[e], (1+p^2)((p^(e+1)-1)/(p^2-1)), 1+2*(p^(e+2)-p^2)/(p^2-1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 53] (* Amiram Eldar, Sep 05 2020 *)

Prog

(PARI) a(n)=sumdiv(n, d, d*core(d)) \\ Charles R Greathouse IV, Oct 30 2011

Crossrefs

Similar to A068976 (sum of square part of d) and A069088 (sum of squarefree part of d).

Cf. A007913, A053143.

Sequence in context: A141622 A144136 A343852 * A001483 A173679 A230208

Adjacent sequences: A198283 A198284 A198285 * A198287 A198288 A198289

Keyword

nonn,mult

Author

Antonio Roldán, Oct 23 2011

Status

approved