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A360162 a(n) is the sum of the square roots of the unitary divisors of n that are squares. 3
1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 5, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 5, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 9, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 5, 10, 1, 1, 3, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
The number of unitary divisors of n that are squares is A056624(n) and their sum is A358347(n).
The unitary analog of A069290.
LINKS
FORMULA
a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} sqrt(d).
Multiplicative with a(p^e) = p^(e/2) + 1 if e is even, and 1 if e is odd.
Dirichlet g.f.: zeta(s)*zeta(2*s-1)/zeta(3*s-1).
Sum_{k=1..n} a(k) ~ (3*n/Pi^2)*(log(n) + 3*gamma - 1 - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).
MATHEMATICA
f[p_, e_] := If[OddQ[e], 1, p^(e/2) + 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, 1, f[i, 1]^(f[i, 2]/2) + 1)); }
CROSSREFS
Sequence in context: A016465 A122947 A367990 * A367988 A363925 A231147
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Jan 29 2023
STATUS
approved

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Last modified April 12 07:22 EDT 2024. Contains 371623 sequences. (Running on oeis4.)