OFFSET
1,3
COMMENTS
In general, for k>=1, Sum_{m=1..n} phi(m)^k ~ c(k) * n^(k+1) / (k+1).
Table of the first twenty constants c(k):
c1 = 0.6079271018540266286632767792583658334261526480334792930736...
c2 = 0.4282495056770944402187657075818235461212985133559361440319...
c3 = 0.3371878737915899719616928161521582449491541277581639388802...
c4 = 0.2862564715115608911732883400866386479560747005250468681615...
c5 = 0.2550316684059564308661179534476184539887434047229867871927...
c6 = 0.2342690874743831026992085481001750961630443094403694748409...
c7 = 0.2194845388428573186801010214226853865762414525869501954550...
c8 = 0.2083553180392308846240883587603960475166426933863125773262...
c9 = 0.1996016550942289223053750541784521301740825495040856984950...
c10 = 0.1924764951305819663569723926235916851341834741671794581256...
c11 = 0.1865198318046079731059147989571847359151227252097897755685...
c12 = 0.1814343147960482243026212589426877406632573154701351352790...
c13 = 0.1770192204728143035012153190352692532613146649385520287635...
c14 = 0.1731338036872585521607716180505314246174563305338731073703...
c15 = 0.1696760784770144194638735708052066949428247152918280392147...
c16 = 0.1665700322333281768929516390245288052095235102037486400080...
c17 = 0.1637576294807392765019551841269187995536332906534705685240...
c18 = 0.1611936368897236567526886186599877745065426644021588804182...
c19 = 0.1588421683609925408830108209202958349394621277940566066627...
c20 = 0.1566743130878534775247182243921577941535243896576096188342...
Conjecture: c(k)*log(k) converges to a constant (around 0.534).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of c(k)*log(k), for k = 1..350
FORMULA
Multiplicative with a(p^e) = (p-1)^4 * p^(4*e-4).
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)).
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...
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
MATHEMATICA
Table[EulerPhi[n]^4, {n, 1, 50}]
PROG
(PARI) a(n) = eulerphi(n)^4;
(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], ", "))
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Vaclav Kotesovec, Mar 02 2023
STATUS
approved