cp's OEIS Frontend

A344222 a(n) = Sum_{k=1..n} tau(gcd(k,n)^4), where tau(n) is the number of divisors of n.

%I A344222 #21 Nov 16 2022 06:56:17
%S A344222 1,6,7,16,9,42,11,36,25,54,15,112,17,66,63,76,21,150,23,144,77,90,27,
%T A344222 252,49,102,79,176,33,378,35,156,105,126,99,400,41,138,119,324,45,462,
%U A344222 47,240,225,162,51,532,81,294,147,272,57,474,135,396,161,198,63,1008,65,210,275,316,153,630,71
%N A344222 a(n) = Sum_{k=1..n} tau(gcd(k,n)^4), where tau(n) is the number of divisors of n.
%H A344222 Seiichi Manyama, <a href="/A344222/b344222.txt">Table of n, a(n) for n = 1..10000</a>
%F A344222 a(n) = Sum_{d|n} phi(n/d) * tau(d^4).
%F A344222 a(n) = n * Sum_{d|n} 4^omega(d) / d.
%F A344222 If p is prime, a(p) = 4 + p.
%F A344222 From _Amiram Eldar_, Nov 15 2022: (Start)
%F A344222 Multiplicative with a(p^e) = (p^e*(p + 3) - 4)/(p - 1).
%F A344222 Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 3/p^2) = 2.4997873122... . (End)
%t A344222 Table[Sum[DivisorSigma[0,GCD[k,n]^4],{k,n}],{n,100}] (* _Giorgos Kalogeropoulos_, May 13 2021 *)
%t A344222 f[p_, e_] := (p^e*(p + 3) - 4)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 15 2022 *)
%o A344222 (PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n)^4));
%o A344222 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^4));
%o A344222 (PARI) a(n) = n*sumdiv(n, d, 4^omega(d)/d);
%Y A344222 Cf. A060648, A072691, A344221, A344223, A344224, A344225.
%K A344222 nonn,mult
%O A344222 1,2
%A A344222 _Seiichi Manyama_, May 12 2021