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A048250
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Sum of squarefree divisors of n.
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24
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1, 3, 4, 3, 6, 12, 8, 3, 4, 18, 12, 12, 14, 24, 24, 3, 18, 12, 20, 18, 32, 36, 24, 12, 6, 42, 4, 24, 30, 72, 32, 3, 48, 54, 48, 12, 38, 60, 56, 18, 42, 96, 44, 36, 24, 72, 48, 12, 8, 18, 72, 42, 54, 12, 72, 24, 80, 90, 60, 72, 62, 96, 32, 3, 84, 144, 68, 54, 96, 144, 72, 12, 74
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OFFSET
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1,2
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COMMENTS
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Also sum of divisors of the squarefree kernel of n: a(n)=A000203(A007947(n)). - Reinhard Zumkeller, Jul 19 2002
Apparently the absolute values of the Dirichlet inverse of A001615. - R. J. Mathar, Dec 22 2010
Row sums of the triangle in A206778. [Reinhard Zumkeller, Feb 12 2012]
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REFERENCES
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D. Suryanarayana, On the core of an integer, Indian J. Math. 14 (1972) 65-74.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
S. R. Finch, Unitarism and infinitarism.
Index entries for sequences related to sums of squares
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FORMULA
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If n = Product p_i^e_i, a(n) = Product (p_i + 1). - Vladeta Jovovic, Apr 19 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(2*s-2). - Michael Somos, Sep 08, 2002.
a(n)=sum(d|n, mu(d)^2*d) - Benoit Cloitre, Dec 09 2002
Pieter Moree (moree(AT)mpim-bonn.mpg.de), Feb 20, 2004 can show that Sum_{n <= x} a(n) = x^2/2 + O(x*sqrt{x}) and adds: "As S. R. Finch pointed out to me, in Suryanarayana's paper this is proved under the Riemann hypothesis with error term O(x^{7/5+\epsilon})".
a(n)=psi(rad(n))=A001615(A007947(n)) [From Enrique Perez Herrero, Aug 24 2010]
a(n)=rad(n)*psi(n)/n=A001615(n)*A007947(n)/n [From Enrique Perez Herrero, Aug 31 2010]
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EXAMPLE
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For n=1000 out of the 16 divisors four are squarefree: {1,2,5,10}. Their sum is 18. Or, 1000=2^3*5^3 hence a(1000)=(2+1)*(5+1)=18.
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MAPLE
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A048250 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ] [ 1 ]): od: RETURN(ans) end:
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MATHEMATICA
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sumOfSquareFreeDivisors[ n_ ] := Plus @@ Select[ Divisors[ n ], MoebiusMu[ # ] != 0 & ]; Table[ sumOfSquareFreeDivisors[ i ], {i, 85} ]
Table[Total[Select[Divisors[n], SquareFreeQ]], {n, 80}] (* Harvey P. Dale, Jan 25 2013 *)
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PROG
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(PARI) a(n)=if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)))
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, (1+p*X)/(1-X))[n])
(PARI) a(n)=sumdiv(n, d, moebius(d)^2*d); /* Joerg Arndt, Jul 06 2011 */
(Haskell)
a034448 = sum . a206778_row -- Reinhard Zumkeller, Feb 12 2012
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CROSSREFS
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Cf. A034444, A034448, A007947, A003557, A023900.
Cf. A206787.
Sequence in context: A101684 A061800 A218789 * A073181 A183100 A046897
Adjacent sequences: A048247 A048248 A048249 * A048251 A048252 A048253
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KEYWORD
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nonn,easy,nice,mult,changed
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu)
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STATUS
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approved
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