A332696 Sum of the proper divisors of n such that d, n/d and n-d are all squarefree.

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

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

Cf. A000005, A008683, A334368.

Sequence in context: A210444 A226949 A166589 * A360880 A255330 A291940

Adjacent sequences: A332693 A332694 A332695 * A332697 A332698 A332699

Keyword

nonn,easy,look

Author

Wesley Ivan Hurt, Apr 26 2020

Status

approved