A029939 a(n) = Sum_{d|n} phi(d)^2.
1, 2, 5, 6, 17, 10, 37, 22, 41, 34, 101, 30, 145, 74, 85, 86, 257, 82, 325, 102, 185, 202, 485, 110, 417, 290, 365, 222, 785, 170, 901, 342, 505, 514, 629, 246, 1297, 650, 725, 374, 1601, 370, 1765, 606, 697, 970, 2117, 430, 1801, 834, 1285, 870, 2705, 730, 1717, 814, 1625
Offset: 1
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
with(numtheory): A029939 := proc(n) local i,j; j := 0; for i in divisors(n) do j := j+phi(i)^2; od; j; end; # alternative N:= 1000: # to get a(1)..a(N) A:= Vector(N,1): for d from 2 to N do pd:= numtheory:-phi(d)^2; md:= [seq(i,i=d..N,d)]; A[md]:= map(`+`,A[md],pd); od: seq(A[i],i=1..N); # Robert Israel, May 30 2016
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Mathematica
Table[Total[EulerPhi[Divisors[n]]^2],{n,60}] (* Harvey P. Dale, Feb 04 2017 *) 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 *) -
PARI
a(n) = sumdiv(n, d, eulerphi(d)^2); \\ Michel Marcus, Jan 17 2017
Formula
Multiplicative with a(p^e) = (p^(2*e)*(p-1)+2)/(p+1). - Vladeta Jovovic, Nov 19 2001
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
a(n) = Sum_{k=1..n} phi(n/gcd(n, k)). - Ridouane Oudra, Nov 28 2019
Sum_{k>=1} 1/a(k) = 2.3943802654751092440350752246012273573942903149891228695146514601814537713... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/(3*zeta(2))) * Product_{p prime} (1 - 1/(p*(p+1))) = A253905 * A065463 / 3 = 0.171593... . - Amiram Eldar, Oct 25 2022
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