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A198286 a(n) = Sum_{d|n} (A053143(d) or smallest square divisible by d). 1
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|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

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Last modified January 29 16:07 EST 2023. Contains 359923 sequences. (Running on oeis4.)