OFFSET
1,4
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
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).
a(p^k) = p^(k-1) * mu(p-1)^2 for k = 1 or 2, and 0 for k > 2.
If p is an odd prime, a(2*p) = p + mu(2*p-1)^2. - Robert Israel, Apr 28 2020
EXAMPLE
a(41) = 0; There are no such divisors of 41 since 1 and 41 are squarefree, but 41 - 1 = 40 is not.
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.
a(43) = 1; The only divisor of 43 that meets all three conditions is 1.
a(44) = 24; The two divisors of 44 that meet all three conditions are 2 and 22. The sum is 2 + 22 = 24.
MAPLE
f:= proc(n) uses numtheory;
convert(select(t-> issqrfree(t) and issqrfree(n/t) and issqrfree(n-t), divisors(n) minus {n}), `+`)
end proc:
map(f, [$1..100]); # Robert Israel, Apr 28 2020
MATHEMATICA
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}]
PROG
(PARI) a(n) = sumdiv(n, d, if ((d!=n) && issquarefree(d) && issquarefree(n/d) && issquarefree(n-d), d)); \\ Michel Marcus, Apr 26 2020
CROSSREFS
KEYWORD
AUTHOR
Wesley Ivan Hurt, Apr 26 2020
STATUS
approved