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A367991
The sum of the divisors of the squarefree part of n.
2
1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72
OFFSET
1,2
COMMENTS
First differs from A348503 at n = 72 and from A344695 at n = 108.
LINKS
FORMULA
Multiplicative with a(p^e) = p + 1 if e is odd and 1 otherwise.
a(n) = A000203(A007913(n)) = A048250(A007913(n)).
a(n) = A048250(n)/A367990(n).
a(n) >= 1, with equality if and only if n is a square (A000290).
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4)/zeta(3) = 0.900392677639... .
MATHEMATICA
f[p_, e_] := If[OddQ[e], p + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1]+1, 1)); }
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 07 2023
STATUS
approved