A332696 - OEIS

%I #52 Apr 28 2020 13:01:10

%S 0,1,1,2,0,4,1,0,3,5,1,8,0,8,6,0,0,3,0,10,7,12,1,0,0,13,0,16,0,16,1,0,

%T 14,18,6,6,0,20,14,0,0,32,1,24,18,23,1,0,7,0,17,26,0,0,0,0,19,30,1,32,

%U 0,32,21,0,0,45,1,36,26,41,1,0,0,38,5,40,18,53,1,0,0,41

%N Sum of the proper divisors of n such that d, n/d and n-d are all squarefree.

%H Robert Israel, <a href="/A332696/b332696.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n, d<n} d * mu(d)^2 * mu(n/d)^2 * mu(n-d)^2, where mu is the Möebius function (A008683).

%F a(p^k) = p^(k-1) * mu(p-1)^2 for k = 1 or 2, and 0 for k > 2.

%F If p is an odd prime, a(2*p) = p + mu(2*p-1)^2. - _Robert Israel_, Apr 28 2020

%e a(41) = 0; There are no such divisors of 41 since 1 and 41 are squarefree, but 41 - 1 = 40 is not.

%e a(42) = 32; The four divisors of 42 that meet all three conditions are 1, 3, 7 and 21. The sum is 1 + 3 + 7 + 21 = 32.

%e a(43) = 1; The only divisor of 43 that meets all three conditions is 1.

%e a(44) = 24; The two divisors of 44 that meet all three conditions are 2 and 22. The sum is 2 + 22 = 24.

%p f:= proc(n) uses numtheory;

%p   convert(select(t-> issqrfree(t) and issqrfree(n/t) and issqrfree(n-t), divisors(n) minus {n}),`+`)

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Apr 28 2020

%t Table[Sum[i*MoebiusMu[i]^2 MoebiusMu[n/i]^2 MoebiusMu[n - i]^2 (1 - Ceiling[n/i] + Floor[n/i]), {i, Floor[n/2]}], {n, 100}]

%o (PARI) a(n) = sumdiv(n, d, if ((d!=n) && issquarefree(d) && issquarefree(n/d) && issquarefree(n-d), d)); \\ _Michel Marcus_, Apr 26 2020

%Y Cf. A000005, A008683, A334368.

%K nonn,easy,look

%O 1,4

%A _Wesley Ivan Hurt_, Apr 26 2020