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A370681
a(n) is the alternating sum of the unitary divisors of n, when these divisors are starting with n and decreasing.
3
1, 1, 2, 3, 4, 4, 6, 7, 8, 6, 10, 10, 12, 8, 12, 15, 16, 10, 18, 18, 16, 12, 22, 18, 24, 14, 26, 24, 28, 22, 30, 31, 24, 18, 32, 30, 36, 20, 28, 36, 40, 32, 42, 36, 40, 24, 46, 34, 48, 26, 36, 42, 52, 28, 48, 54, 40, 30, 58, 46, 60, 32, 60, 63, 56, 48, 66, 54
OFFSET
1,3
COMMENTS
a(n) is odd if and only if n is a power of 2.
LINKS
FORMULA
a(n) = A071324(n) if n is squarefree (A005117) or if n is in A370683.
EXAMPLE
The unitary divisors of 6 are {1, 2, 3, 6}, hence a(6) = 6 - 3 + 2 - 1 = 4.
The unitary divisors of 12 are {1, 3, 4, 12}, hence a(12) = 12 - 4 + 3 - 1 = 10.
MATHEMATICA
a[n_] := Module[{d = Reverse[Select[Divisors[n], CoprimeQ[#, n/#] &]]}, Total[(-1)^(Range[Length[d]] + 1)*d]]; Array[a, 100]
PROG
(PARI) a(n) = {my(d = Vecrev(select(x->(gcd(x, n/x) == 1), divisors(n)))); sum(i=1, #d, (-1)^(i+1)*d[i]); }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Feb 26 2024
STATUS
approved