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A332696 Sum of the proper divisors of n such that d, n/d and n-d are all squarefree. 2
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 (list; graph; refs; listen; history; text; internal format)
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 * A255330 A291940 A153345

Adjacent sequences:  A332693 A332694 A332695 * A332697 A332698 A332699

KEYWORD

nonn,easy,look

AUTHOR

Wesley Ivan Hurt, Apr 26 2020

STATUS

approved

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Last modified October 17 15:30 EDT 2021. Contains 348063 sequences. (Running on oeis4.)