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A198286 Sum( d divides n, A053143(d), or smallest square divisible by d) 0
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 (list; graph; refs; listen; history; text; internal format)
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 divides n, d*core(d)) where core(x) is the squarefree part of x), i.e., inverse Mobius transform of A053143.

LINKS

Table of n, a(n) for n=1..53.

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, 20=2^2*5, 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 *)

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).

Sequence in context: A109360 A141622 A144136 * A001483 A173679 A230208

Adjacent sequences:  A198283 A198284 A198285 * A198287 A198288 A198289

KEYWORD

nonn,mult

AUTHOR

Antonio Roldán, Oct 23 2011

STATUS

approved

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Last modified December 7 17:41 EST 2019. Contains 329847 sequences. (Running on oeis4.)