A327171 a(n) = phi(n) * core(n), where phi is Euler totient function, and core gives the squarefree part of n.

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

Antti Karttunen, Table of n, a(n) for n = 1..20000

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. A000010, A007913, A053143, A248003, A327170, A327172.

Cf. also A002618, A062355.

Sequence in context: A174857 A248568 A257252 * A008556 A254638 A320118

Adjacent sequences: A327168 A327169 A327170 * A327172 A327173 A327174

Keyword

nonn,easy,mult

Author

Antti Karttunen, Sep 28 2019

Status

approved