%I #18 Sep 01 2023 04:09:17

%S 1,1,16,16,256,16,1296,256,1296,256,10000,256,20736,1296,4096,4096,

%T 65536,1296,104976,4096,20736,10000,234256,4096,160000,20736,104976,

%U 20736,614656,4096,810000,65536,160000,65536,331776,20736,1679616,104976,331776,65536,2560000

%N a(n) = phi(n)^4.

%C In general, for k>=1, Sum_{m=1..n} phi(m)^k ~ c(k) * n^(k+1) / (k+1).

%C Table of the first twenty constants c(k):

%C c1 = 0.6079271018540266286632767792583658334261526480334792930736...

%C c2 = 0.4282495056770944402187657075818235461212985133559361440319...

%C c3 = 0.3371878737915899719616928161521582449491541277581639388802...

%C c4 = 0.2862564715115608911732883400866386479560747005250468681615...

%C c5 = 0.2550316684059564308661179534476184539887434047229867871927...

%C c6 = 0.2342690874743831026992085481001750961630443094403694748409...

%C c7 = 0.2194845388428573186801010214226853865762414525869501954550...

%C c8 = 0.2083553180392308846240883587603960475166426933863125773262...

%C c9 = 0.1996016550942289223053750541784521301740825495040856984950...

%C c10 = 0.1924764951305819663569723926235916851341834741671794581256...

%C c11 = 0.1865198318046079731059147989571847359151227252097897755685...

%C c12 = 0.1814343147960482243026212589426877406632573154701351352790...

%C c13 = 0.1770192204728143035012153190352692532613146649385520287635...

%C c14 = 0.1731338036872585521607716180505314246174563305338731073703...

%C c15 = 0.1696760784770144194638735708052066949428247152918280392147...

%C c16 = 0.1665700322333281768929516390245288052095235102037486400080...

%C c17 = 0.1637576294807392765019551841269187995536332906534705685240...

%C c18 = 0.1611936368897236567526886186599877745065426644021588804182...

%C c19 = 0.1588421683609925408830108209202958349394621277940566066627...

%C c20 = 0.1566743130878534775247182243921577941535243896576096188342...

%C c1 = A059956 = 6/Pi^2, c2 = A065464.

%C Conjecture: c(k)*log(k) converges to a constant (around 0.534).

%H Amiram Eldar, <a href="/A361148/b361148.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A361148/a361148.jpg">Plot of c(k)*log(k), for k = 1..350</a>

%F Multiplicative with a(p^e) = (p-1)^4 * p^(4*e-4).

%F Dirichlet g.f.: zeta(s-4) * Product_{primes p} (1 + 1/p^s - 4/p^(s-1) + 6/p^(s-2) - 4/p^(s-3)).

%F Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = Product_{primes p} (1 - 4/p^2 + 6/p^3 - 4/p^4 + 1/p^5) = 0.286256471511560891173288340086638647956074700525046868161...

%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p^4/((p-1)^4*(p^4-1))) = 2.20815077889083518654... . - _Amiram Eldar_, Sep 01 2023

%t Table[EulerPhi[n]^4, {n, 1, 50}]

%o (PARI) a(n) = eulerphi(n)^4;

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X - 4*p*X + 6*p^2*X - 4*p^3*X) / (1 - p^4*X))[n], ", "))

%Y Cf. A000010, A002088, A127473, A057434, A358714, A361132, A361179.

%Y Cf. A059956, A065464.

%K nonn,easy,mult

%O 1,3

%A _Vaclav Kotesovec_, Mar 02 2023