A327171 - OEIS

%I #36 Jul 13 2024 04:03:11

%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

%o (Magma) [EulerPhi(n)*Squarefree(n): n in [1..100]]; // _G. C. Greubel_, Jul 13 2024

%o (SageMath) [euler_phi(n)*squarefree_part(n) for n in range(1,101)] # _G. C. Greubel_, Jul 13 2024

%Y Cf. A082473 (gives the terms in ascending order, with duplicates removed).

%Y Cf. A000010, A007913, A053143, A248003, A327170, A327172.

%Y Cf. also A002618, A062355.

%K nonn,easy,mult

%O 1,2

%A _Antti Karttunen_, Sep 28 2019