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%I #62 Mar 10 2023 12:21:07
%S 1,1,4,4,16,4,36,16,36,16,100,16,144,36,64,64,256,36,324,64,144,100,
%T 484,64,400,144,324,144,784,64,900,256,400,256,576,144,1296,324,576,
%U 256,1600,144,1764,400,576,484,2116,256,1764,400,1024,576,2704,324,1600
%N a(n) = phi(n)^2.
%C Number of maps of the form j |--> m*j + d with gcd(m, n) = 1 and gcd(d, n) = 1 from [1, 2, ..., n] to itself. - _Joerg Arndt_, Aug 29 2014
%C Right border of A127474.
%C Equals the Mobius transform (A054525) of A029939. - _Gary W. Adamson_, Aug 20 2008
%C From _Jianing Song_, Apr 14 2019: (Start)
%C a(n) is the number of solutions to gcd(xy, n) = 1 with x, y in [0, n-1].
%C Let Z_n be the ring of integers modulo n, then a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - x) (or equivalently, Z_n[x]/(x^2 + x)) with discriminant d = 1 (that is, a(n) is the size of the group G(n) = (Z_n[x]/(x^2 - x))*). Actually, G(n) is isomorphic to (Z_n)* X (Z_n)*. (End)
%H Jens Kruse Andersen, <a href="/A127473/b127473.txt">Table of n, a(n) for n = 1..10000</a>
%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence).
%F a(n) = A000010(n)^2.
%F Multiplicative with a(p^e) = (p-1)^2*p^(2e-2), e >= 1. Dirichlet g.f. zeta(s-2)*Product_{primes p} (1 - 2/p^(s-1) + 1/p^s). - _R. J. Mathar_, Apr 04 2011
%F Sum_{k>=1} 1/a(k) = A109695. - _Vaclav Kotesovec_, Sep 20 2020
%F Sum_{k>=1} (-1)^k/a(k) = (1/7) * A109695. - _Amiram Eldar_, Nov 11 2020
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime}(1 - (2*p-1)/p^3) = A065464 / 3 = 0.142749... . - _Amiram Eldar_, Oct 25 2022
%e a(5) = 16 since phi(5) = 4.
%p A127473 := proc(n) numtheory[phi](n)^2 ; end proc:
%p seq(A127473(n),n=1..40) ; # _R. J. Mathar_, Apr 04 2011
%t Table[EulerPhi[n]^2,{n,100}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 02 2012 *)
%o (Magma) [(EulerPhi(n))^2: n in [1..180]]; // _Vincenzo Librandi_, Apr 04 2011
%o (PARI) a(n) = eulerphi(n)^2; \\ _Michel Marcus_, Oct 16 2018
%Y Cf. A000010, A057434, A065464, A109695, A127474, A358714, A361148.
%Y Similar sequences: A082953 (size of (Z_n[x]/(x^2 - 1))*, d = 4), A002618 ((Z_n[x]/(x^2))*, d = 0), A079458 ((Z_n[x]/(x^2 + 1))*, d = -4), A319445 ((Z_n[x]/(x^2 - x + 1))* or (Z_n[x]/(x^2 + x + 1))*, d = -3).
%K nonn,mult
%O 1,3
%A _Gary W. Adamson_, Jan 15 2007
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