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a(n) = Sum_{d|n} phi(d) * sigma(d) * tau(n/d).
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%I #21 Jan 26 2023 03:53:27

%S 1,5,10,23,26,50,50,101,97,130,122,230,170,250,260,427,290,485,362,

%T 598,500,610,530,1010,671,850,904,1150,842,1300,962,1761,1220,1450,

%U 1300,2231,1370,1810,1700,2626,1682,2500,1850,2806,2522,2650,2210,4270,2493,3355,2900,3910,2810,4520

%N a(n) = Sum_{d|n} phi(d) * sigma(d) * tau(n/d).

%C Inverse Moebius transform of A062952.

%H Michael De Vlieger, <a href="/A341638/b341638.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n} tau(gcd(n,k)) * sigma(n/gcd(n,k)).

%F a(n) = Sum_{d|n} A062952(d).

%F a(n) = Sum_{k=1..n} tau(n/gcd(n,k))*sigma(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - _Richard L. Ollerton_, May 09 2021

%F From _Amiram Eldar_, Jan 26 2023: (Start)

%F Multiplicative with a(p^e) = (p^(2*e+4) - p^(e+3) - 2*p^(e+2) - p^(e+1) + (e+1)*p^3 - (e-1)*p + 1)/(p^2-1)^2.

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)^2/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A013661 * A002117^2 * A330523 / 3 = 0.424578... . (End)

%t Table[Sum[EulerPhi[d] DivisorSigma[1, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 54}]

%t Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[1, n/GCD[n, k]], {k, n}], {n, 54}]

%t f[p_, e_] := (p^(2*e + 4) - p^(e + 3) - 2*p^(e + 2) - p^(e + 1) + (e + 1)*p^3 - (e - 1)*p + 1)/(p^2 - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 26 2023 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)*numdiv(n/d)); \\ _Michel Marcus_, Feb 17 2021

%Y Cf. A000005, A000010, A000203, A007430, A062354, A062952, A191831.

%Y Cf. A002117, A013661, A330523.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Feb 16 2021