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%I #32 Sep 15 2023 06:36:30
%S 1,2,6,2,20,12,42,8,6,40,110,12,156,84,120,8,272,12,342,40,252,220,
%T 506,48,20,312,54,84,812,240,930,32,660,544,840,12,1332,684,936,160,
%U 1640,504,1806,220,120,1012,2162,48,42,40,1632,312,2756,108,2200,336,2052,1624,3422,240,3660,1860,252,32,3120,1320
%N a(n) = phi(n) * core(n), where phi is Euler totient function, and core gives the squarefree part of n.
%D Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 161.
%H Antti Karttunen, <a href="/A327171/b327171.txt">Table of n, a(n) for n = 1..20000</a>
%H Chantal David and Francesco Pappalardi, <a href="https://doi.org/10.1155/S1073792899000082">Average Frobenius distributions of elliptic curves</a>, International Mathematics Research Notices, Vol. 1999, No. 4 (1999), pp. 165-183, <a href="http://web2.mat.uniroma3.it/users/pappa/papers/IMRN_4_1999.pdf">alternative link</a>.
%F a(n) = A000010(n) * A007913(n).
%F Multiplicative with a(p^k) = (p-1) * p^((k-1)+(k mod 2)).
%F Sum_{n>=1} 1/a(n) = (Pi^2/6) * Product_{p prime} (1 + (p+1)/(p^2*(p-1))) = 3.96555686901754604330... - _Amiram Eldar_, Oct 16 2020
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/45) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.1500809164... . - _Amiram Eldar_, Dec 05 2022
%F a(n) = A000010(A053143(n)). - _Amiram Eldar_, Sep 15 2023
%t Array[EulerPhi[#] (Sqrt@ # /. (c_: 1) a_^(b_: 0) :> (c a^b)^2) &, 66] (* _Michael De Vlieger_, Sep 29 2019, after _Bill Gosper_ at A007913 *)
%o (PARI) A327171(n) = eulerphi(n)*core(n);
%o (PARI) A327171(n) = { my(f=factor(n)); prod (i=1, #f~, (f[i, 1]-1)*(f[i, 1]^(-1 + f[i, 2] + (f[i, 2]%2)))); };
%o (Python)
%o from sympy.ntheory.factor_ import totient, core
%o def A327171(n):
%o return totient(n)*core(n) # _Chai Wah Wu_, Sep 29 2019
%Y Cf. A082473 (gives the terms in ascending order, with duplicates removed).
%Y Cf. A000010, A007913, A053143, A327170, A327172, A248003.
%Y Cf. also A002618, A062355.
%K nonn,easy,mult
%O 1,2
%A _Antti Karttunen_, Sep 28 2019
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