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A358346
a(n) is the sum of the unitary divisors of n that are exponentially odd (A268335).
4
1, 3, 4, 1, 6, 12, 8, 9, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 36, 1, 42, 28, 8, 30, 72, 32, 33, 48, 54, 48, 1, 38, 60, 56, 54, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 84, 72, 72, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144
OFFSET
1,2
COMMENTS
The number of unitary divisors of n that are exponentially odd is A055076(n).
LINKS
FORMULA
a(n) >= 1 with equality if and only if n is a square (A000290).
a(n) <= A033634(n) with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^e + 1 if e is odd, and 1 otherwise.
a(n) = A034448(n)/A358347(n).
Sum_{k=1..n} a(k) ~ n^2/2.
From Amiram Eldar, Sep 14 2023: (Start)
a(n) = A034448(A350389(n)).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(2*s-1)). (End)
MATHEMATICA
f[p_, e_] := 1 + If[OddQ[e], p^e, 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i, 2]%2, f[i, 1]^f[i, 2], 0)); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Nov 11 2022
STATUS
approved