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%I #32 Jan 22 2024 13:02:53
%S 1,9,55,137,501,495,2059,2185,4429,4509,13311,7535,26365,18531,27555,
%T 34953,78609,39861,123463,68637,113245,119799,267675,120175,313001,
%U 237285,358723,282083,682893,247995,893731,559241,732105,707481,1031559,606773,1823509
%N a(n) = Sum_{d|n} phi(d^4).
%H Robert Israel, <a href="/A068970/b068970.txt">Table of n, a(n) for n = 1..10000</a>
%F Also Sum_{d|n} d^m*phi(d^(4-m)) for m=0, 1, 2, 3.
%F Multiplicative with a(p^e) = 1 + p^3 * (p-1)(p^(4e)-1)/(p^4-1).
%F G.f.: Sum_{k>=1} k^3*phi(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Mar 10 2018
%F Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(2)) = 0.126075... . - _Amiram Eldar_, Dec 01 2022
%F From _Peter Bala_, Jan 21 2024: (Start)
%F a(n) = Sum_{k = 1..n} (n/gcd(k, n))^3 = Sum_{k = 1..n} (lcm(k, n)/k)^3.
%F Dirichlet g.f.: zeta(s) * zeta(s-4)/zeta(s-3). (End)
%p f:= n -> add(numtheory:-phi(d^4),d=numtheory:-divisors(n)):
%p map(f, [$1..100]); # _Robert Israel_, Sep 13 2018
%t Table[Total[EulerPhi[Divisors[n]^4]], {n, 40}] (* _Vincenzo Librandi_, Sep 13 2018 *)
%t f[p_, e_] := 1 + p^3*(p - 1)*(p^(4*e) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* _Amiram Eldar_, Dec 01 2022 *)
%o (PARI) a(n) = sumdiv(n, d, eulerphi(d^4)); \\ _Michel Marcus_, Mar 10 2018
%Y Cf. A000010, A013661, A013663, A057660, A068963, A189393.
%K easy,nonn,mult
%O 1,2
%A _Benoit Cloitre_, Apr 06 2002
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