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A053164
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4th root of largest 4th power dividing n.
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21
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2
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OFFSET
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1,16
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COMMENTS
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Multiplicative with a(p^e) = p^[e/4]. - Mitch Harris, Apr 19 2005
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LINKS
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FORMULA
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Multiplicative with a(p^e) = p^[e/4].
Dirichlet g.f.: zeta(4s-1)*zeta(s)/zeta(4s). - R. J. Mathar, Apr 09 2011
Sum_{k=1..n} a(k) ~ 90*zeta(3)*n/Pi^4 + 3*zeta(1/2)*sqrt(n)/Pi^2. - Vaclav Kotesovec, Dec 01 2020
G.f.: Sum_{k>=1} phi(k) * x^(k^4) / (1 - x^(k^4)). - Ilya Gutkovskiy, Aug 20 2021
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EXAMPLE
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a(32) = 2 since 2 = 16^(1/4) and 16 is the largest 4th power dividing 32.
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MAPLE
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A053164 := proc(n) local a, f, e, p ; for f in ifactors(n)[2] do e:= op(2, f) ; p := op(1, f) ; a := a*p^floor(e/4) ; end do ; a ; end proc: # R. J. Mathar, Jan 11 2012
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MATHEMATICA
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f[list_] := list[[1]]^Quotient[list[[2]], 4]; Table[Apply[Times, Map[f, FactorInteger[n]]], {n, 1, 81}] (* Geoffrey Critzer, Jan 21 2015 *)
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PROG
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(Scheme, with memoization macro definec)
(define (A002265 n) (floor->exact (/ n 4))) ;; For MIT/GNU Scheme
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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