A372999 - OEIS

%I #14 May 22 2024 02:00:05

%S 1,6,11,24,21,66,31,76,65,126,51,264,61,186,231,212,81,390,91,504,341,

%T 306,111,836,201,366,299,744,141,1386,151,548,561,486,651,1560,181,

%U 546,671,1596,201,2046,211,1224,1365,666,231,2332,409,1206,891,1464,261,1794

%N a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^4 ).

%H Amiram Eldar, <a href="/A372999/b372999.txt">Table of n, a(n) for n = 1..10000</a>

%F If p is prime, a(p) = 5*p - 4.

%F a(n) = Sum_{d|n} phi(d) * tau(d^4).

%F Multiplicative with a(p^e) = (4 - (4*e+5)*p^e + (4*e+1)*p^(e+1))/(p-1). - _Amiram Eldar_, May 21 2024

%t f[p_, e_] := (4 - (4*e+5)*p^e + (4*e+1)*p^(e+1))/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(d)*numdiv(d^4));

%Y Cf. A062380, A062949, A372997.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, May 19 2024