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A319072
a(n) is the sum of the non-bi-unitary divisors of n.
2
0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 0, 4, 0, 9, 0, 12, 0, 0, 0, 0, 5, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 24, 18, 0, 0, 16, 7, 15, 0, 28, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 24, 8, 0, 0, 0, 36, 0, 0, 0, 45, 0, 0, 20, 40, 0, 0, 0, 24, 9, 0, 0, 64, 0, 0, 0, 0, 0, 54, 0, 48, 0, 0, 0, 0, 0, 21, 36, 87
OFFSET
1,4
LINKS
FORMULA
a(n) = A000203(n) - A188999(n).
EXAMPLE
For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12, and the bi-unitary divisors of 12 are 1, 3, 4, 12, hence the non-bi-unitary divisors of 12 are 2 and 6, and the sum of them is 2 + 6 = 8, so a(12) = 8. Also the sum of the divisors of 12 is 28, and the sum of the bi-unitary divisors of 12 is 20, so a(12) = 28 - 20 = 8.
MATHEMATICA
f1[p_, e_] := (p^(e+1) - 1)/(p - 1); f2[p_, e_] := f1[p, e] - If[OddQ[e], 0, p^(e/2)]; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100] (* Amiram Eldar, Apr 04 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Sep 22 2018
STATUS
approved