A373103 - OEIS

%I #16 May 26 2024 15:05:12

%S 1,497,19603,254449,1952501,9742691,40351207,130277873,385845769,

%T 970392997,2357933051,4987963747,10604470813,20054549879,38274877103,

%U 66702270961,118587792977,191765347193,322687567459,496811926949,791004710821,1171892726347,1801152381623

%N a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^4.

%H Amiram Eldar, <a href="/A373103/b373103.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.

%F a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_9(d).

%F a(n) = Sum_{d|n} phi(n/d) * (n/d)^8 * sigma_8(d^2)/sigma_4(d^2).

%F From _Amiram Eldar_, May 25 2024: (Start)

%F Multiplicative with a(p^e) = (p^(9*e+9) - p^(9*e+4) + p^4 - 1)/(p^9-1).

%F Dirichlet g.f.: zeta(s)*zeta(s-9)/zeta(s-4).

%F Sum_{k=1..n} a(k) ~ c * n^10 / 10, where c = zeta(10)/zeta(6) = Pi^4/99 = 0.983930212464... . (End)

%t f[p_, e_] :=  (p^(9*e+9) - p^(9*e+4) + p^4 - 1)/(p^9-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* _Amiram Eldar_, May 25 2024 *)

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 9));

%o (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^8*sigma(d^2, 8)/sigma(d^2, 4));

%Y Cf. A371491, A371878, A373007, A373105.

%Y Cf. A350156, A372962, A372964.

%Y Cf. A013664, A013668.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 25 2024