A341636 - OEIS

%I #13 Sep 15 2023 05:53:07

%S 1,4,6,13,10,24,14,38,29,40,22,78,26,56,60,103,34,116,38,130,84,88,46,

%T 228,79,104,124,182,58,240,62,264,132,136,140,377,74,152,156,380,82,

%U 336,86,286,290,184,94,618,153,316,204,338,106,496,220,532,228,232,118,780,122,248

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

%C Inverse Moebius transform of A062949.

%H Amiram Eldar, <a href="/A341636/b341636.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = Sum_{d|n} A062949(d).

%F Multiplicative with a(p^e) = (p + 1 + e*(p-1) + p^(e+1)*(e*(p-1)+p-3))/(p-1)^2. - _Amiram Eldar_, Sep 15 2023

%t Table[Sum[EulerPhi[d] DivisorSigma[0, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 62}]

%t Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[0, n/GCD[n, k]], {k, n}], {n, 62}]

%t f[p_, e_] := (p + 1 + e*(p - 1) + p^(e + 1)*(e*(p - 1) + p - 3))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 15 2023 *)

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

%Y Cf. A000005, A000010, A000203, A007426, A055507, A062355, A062949.

%K nonn,easy,mult

%O 1,2

%A _Ilya Gutkovskiy_, Feb 16 2021