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A327171
a(n) = phi(n) * core(n), where phi is Euler totient function, and core gives the squarefree part of n.
3
1, 2, 6, 2, 20, 12, 42, 8, 6, 40, 110, 12, 156, 84, 120, 8, 272, 12, 342, 40, 252, 220, 506, 48, 20, 312, 54, 84, 812, 240, 930, 32, 660, 544, 840, 12, 1332, 684, 936, 160, 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
OFFSET
1,2
REFERENCES
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 161.
LINKS
Chantal David and Francesco Pappalardi, Average Frobenius distributions of elliptic curves, International Mathematics Research Notices, Vol. 1999, No. 4 (1999), pp. 165-183, alternative link.
FORMULA
a(n) = A000010(n) * A007913(n).
Multiplicative with a(p^k) = (p-1) * p^((k-1)+(k mod 2)).
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
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
a(n) = A000010(A053143(n)). - Amiram Eldar, Sep 15 2023
MATHEMATICA
Array[EulerPhi[#] (Sqrt@ # /. (c_: 1) a_^(b_: 0) :> (c a^b)^2) &, 66] (* Michael De Vlieger, Sep 29 2019, after Bill Gosper at A007913 *)
PROG
(PARI) A327171(n) = eulerphi(n)*core(n);
(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)))); };
(Python)
from sympy.ntheory.factor_ import totient, core
def A327171(n):
return totient(n)*core(n) # Chai Wah Wu, Sep 29 2019
(Magma) [EulerPhi(n)*Squarefree(n): n in [1..100]]; // G. C. Greubel, Jul 13 2024
(SageMath) [euler_phi(n)*squarefree_part(n) for n in range(1, 101)] # G. C. Greubel, Jul 13 2024
CROSSREFS
Cf. A082473 (gives the terms in ascending order, with duplicates removed).
Cf. also A002618, A062355.
Sequence in context: A174857 A248568 A257252 * A008556 A254638 A320118
KEYWORD
nonn,easy,mult
AUTHOR
Antti Karttunen, Sep 28 2019
STATUS
approved