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A048250
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Sum of the squarefree divisors of n.
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133
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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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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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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
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})".
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = 1. - Amiram Eldar, Jun 10 2020
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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:
# alternative:
seq(mul(1+p, p = numtheory:-factorset(n)), n=1..1000); # Robert Israel, Mar 18 2015
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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 *)
a[1] = 1; a[n_] := Times@@(1 + FactorInteger[n][[;; , 1]]); Array[a, 100] (* Amiram Eldar, Dec 19 2018 *)
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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)
(Sage)
def A048250(n): return mul(map(lambda p: p+1, prime_divisors(n)))
(Python)
from math import prod
from sympy import primefactors
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CROSSREFS
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Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), this sequence (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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STATUS
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approved
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