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A303222 Total volume of all rectangular prisms with dimensions p, q and (p + q)/2 such that p and q are squarefree, n = p + q and p <= q.

%I A303222 #14 Sep 08 2022 08:46:21
%S A303222 0,1,3,14,15,42,56,136,144,230,220,612,611,665,675,1576,1768,1836,
%T A303222 1729,4200,3528,4279,3404,7524,6625,8333,8289,14336,11165,12675,10323,
%U A303222 20592,17688,23307,17570,40410,27861,30153,28899,52180,42804,45864,55169,84920
%N A303222 Total volume of all rectangular prisms with dimensions p, q and (p + q)/2 such that p and q are squarefree, n = p + q and p <= q.
%H A303222 Robert Israel, <a href="/A303222/b303222.txt">Table of n, a(n) for n = 1..10000</a>
%H A303222 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A303222 a(n) = (n/2) * Sum_{i=1..floor(n/2)} i * (n-i) * mu(i)^2 * mu(n-i)^2, where mu is the Möbius function (A008683).
%p A303222 N:= 100: # for a(1)..a(N)
%p A303222 A:= Vector(N):
%p A303222 SF:= select(numtheory:-issqrfree, [$1..N-1]):
%p A303222 for iq from 1 to nops(SF) do
%p A303222   q:= SF[iq];
%p A303222   for ip from 1 to iq do
%p A303222     p:= SF[ip];
%p A303222     n:= p+q;
%p A303222     if n > N then break fi;
%p A303222     A[n]:= A[n] + p*q*(p+q)/2
%p A303222   od
%p A303222 od:
%p A303222 convert(A,list); # _Robert Israel_, Jun 12 2018
%t A303222 Table[(n/2)*Sum[i (n - i)*MoebiusMu[i]^2 MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 80}]
%o A303222 (Magma) [0] cat [&+[k*(n-k)*(MoebiusMu(k)^2*MoebiusMu(n-k)^2)*n/2: k in [1..Floor(n/2)]]: n in [2..60]]; // _Vincenzo Librandi_, Apr 21 2018
%o A303222 (PARI) a(n) = n*sum(i=1, n\2, i*(n-i)*moebius(i)^2*moebius(n-i)^2)/2; \\ _Michel Marcus_, Apr 21 2018
%Y A303222 Cf. A008683, A262351.
%K A303222 nonn,easy
%O A303222 1,3
%A A303222 _Wesley Ivan Hurt_, Apr 19 2018