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
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.
Crossrefs
Programs
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Magma
[EulerPhi(n)*Squarefree(n): n in [1..100]]; // G. C. Greubel, Jul 13 2024
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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 *)
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PARI
A327171(n) = eulerphi(n)*core(n);
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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)))); };
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Python
from sympy.ntheory.factor_ import totient, core def A327171(n): return totient(n)*core(n) # Chai Wah Wu, Sep 29 2019
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SageMath
[euler_phi(n)*squarefree_part(n) for n in range(1,101)] # G. C. Greubel, Jul 13 2024
Formula
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