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

Author

Antonio Roldán, Oct 23 2011

Keywords

• nonn
• mult

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.

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A007913, A053143.

Programs

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 *)

PARI

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

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