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A029939 a(n) = Sum_{d|n} phi(d)^2. 10

%I

%S 1,2,5,6,17,10,37,22,41,34,101,30,145,74,85,86,257,82,325,102,185,202,

%T 485,110,417,290,365,222,785,170,901,342,505,514,629,246,1297,650,725,

%U 374,1601,370,1765,606,697,970,2117,430,1801,834,1285,870,2705,730,1717,814,1625

%N a(n) = Sum_{d|n} phi(d)^2.

%C Equals the inverse Mobius transform (A051731) of A127473. [_Gary W. Adamson_, Aug 20 2008]

%H Robert Israel, <a href="/A029939/b029939.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = (p^(2*e)*(p-1)+2)/(p+1). - _Vladeta Jovovic_, Nov 19 2001

%F G.f.: Sum_{k>=1} phi(k)^2*x^k/(1 - x^k), where phi(k) is the Euler totient function (A000010). - _Ilya Gutkovskiy_, Jan 16 2017

%F a(n) = Sum_{k=1..n} phi(n/gcd(n, k)). - _Ridouane Oudra_, Nov 28 2019

%F Sum_{k>=1} 1/a(k) = 2.3943802654751092440350752246012273573942903149891228695146514601814537713... - _Vaclav Kotesovec_, Sep 20 2020

%p with(numtheory): A029939 := proc(n) local i,j; j := 0; for i in divisors(n) do j := j+phi(i)^2; od; j; end;

%p # alternative

%p N:= 1000: # to get a(1)..a(N)

%p A:= Vector(N,1):

%p for d from 2 to N do

%p pd:= numtheory:-phi(d)^2;

%p md:= [seq(i,i=d..N,d)];

%p A[md]:= map(`+`,A[md],pd);

%p od:

%p seq(A[i],i=1..N); # _Robert Israel_, May 30 2016

%t Table[Total[EulerPhi[Divisors[n]]^2],{n,60}] (* _Harvey P. Dale_, Feb 04 2017 *)

%t f[p_, e_] := (p^(2*e)*(p-1)+2)/(p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 18 2020 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(d)^2); \\ _Michel Marcus_, Jan 17 2017

%Y Cf. A051731, A062367, A127473.

%K nonn,mult,easy,changed

%O 1,2

%A _N. J. A. Sloane_

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Last modified September 23 16:41 EDT 2020. Contains 337315 sequences. (Running on oeis4.)