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%I #90 Jan 28 2024 09:39:10
%S 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,
%T 4,24,30,72,32,3,48,54,48,12,38,60,56,18,42,96,44,36,24,72,48,12,8,18,
%U 72,42,54,12,72,24,80,90,60,72,62,96,32,3,84,144,68,54,96,144,72,12,74
%N Sum of the squarefree divisors of n.
%C Also sum of divisors of the squarefree kernel of n: a(n) = A000203(A007947(n)). - _Reinhard Zumkeller_, Jul 19 2002
%C The absolute values of the Dirichlet inverse of A001615. - _R. J. Mathar_, Dec 22 2010
%C Row sums of the triangle in A206778. - _Reinhard Zumkeller_, Feb 12 2012
%C Inverse Möbius transform of n * mu(n)^2 = |A055615(n)|. - _Wesley Ivan Hurt_, Jun 08 2023
%D D. Suryanarayana, On the core of an integer, Indian J. Math. 14 (1972) 65-74.
%H T. D. Noe, <a href="/A048250/b048250.txt">Table of n, a(n) for n = 1..10000</a>
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Unitarism and infinitarism</a>.
%H Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004. [Cached copy, with permission of the author]
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F If n = Product p_i^e_i, a(n) = Product (p_i + 1). - _Vladeta Jovovic_, Apr 19 2001
%F Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(2*s-2). - _Michael Somos_, Sep 08 2002
%F a(n) = Sum_{d|n} mu(d)^2*d = Sum_{d|n} |A055615(d)|. - _Benoit Cloitre_, Dec 09 2002
%F 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})".
%F a(n) = psi(rad(n)) = A001615(A007947(n)). - _Enrique Pérez Herrero_, Aug 24 2010
%F a(n) = rad(n)*psi(n)/n = A001615(n)*A007947(n)/n. - _Enrique Pérez Herrero_, Aug 31 2010
%F G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Jan 03 2017
%F Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = 1. - _Amiram Eldar_, Jun 10 2020
%F a(n) = Sum_{d divides n} mu(d)^2*core(d), where core(n) = A007913(n). - _Peter Bala_, Jan 24 2024
%e 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.
%p 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:
%p # alternative:
%p seq(mul(1+p, p = numtheory:-factorset(n)), n=1..1000); # _Robert Israel_, Mar 18 2015
%t sumOfSquareFreeDivisors[ n_ ] := Plus @@ Select[ Divisors[ n ], MoebiusMu[ # ] != 0 & ]; Table[ sumOfSquareFreeDivisors[ i ], {i, 85} ]
%t Table[Total[Select[Divisors[n],SquareFreeQ]],{n,80}] (* _Harvey P. Dale_, Jan 25 2013 *)
%t a[1] = 1; a[n_] := Times@@(1 + FactorInteger[n][[;;,1]]); Array[a, 100] (* _Amiram Eldar_, Dec 19 2018 *)
%o (PARI) a(n)=if(n<1,0,sumdiv(n,d,if(core(d)==d,d)))
%o (PARI) a(n)=if(n<1,0,direuler(p=2,n,(1+p*X)/(1-X))[n])
%o (PARI) a(n)=sumdiv(n,d,moebius(d)^2*d); \\ _Joerg Arndt_, Jul 06 2011
%o (PARI) a(n)=my(f=factor(n)); for(i=1,#f~,f[i,2]=1); sigma(f) \\ _Charles R Greathouse IV_, Sep 09 2014
%o (Haskell)
%o a034448 = sum . a206778_row -- _Reinhard Zumkeller_, Feb 12 2012
%o (Sage)
%o def A048250(n): return mul(map(lambda p: p+1, prime_divisors(n)))
%o [A048250(n) for n in (1..73)] # _Peter Luschny_, May 23 2013
%o (Python)
%o from math import prod
%o from sympy import primefactors
%o def A048250(n): return prod(p+1 for p in primefactors(n)) # _Chai Wah Wu_, Apr 20 2023
%Y Cf. A003557, A007913, A007947, A023900, A034448, A206787, A309192.
%Y 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).
%Y Cf. A240976 (tenth of Dgf at s=3).
%K nonn,easy,nice,mult
%O 1,2
%A _Labos Elemer_
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