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A053164 4th root of largest 4th power dividing n.

%I A053164 #40 Sep 01 2024 02:23:09
%S A053164 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,
%T A053164 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,
%U A053164 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
%N A053164 4th root of largest 4th power dividing n.
%C A053164 Multiplicative with a(p^e) = p^[e/4]. - _Mitch Harris_, Apr 19 2005
%H A053164 Antti Karttunen, <a href="/A053164/b053164.txt">Table of n, a(n) for n = 1..10000</a>
%H A053164 Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>.
%F A053164 a(n) = A000188(A000188(n)) = A008835(n)^(1/4).
%F A053164 Multiplicative with a(p^e) = p^[e/4].
%F A053164 Dirichlet g.f.: zeta(4s-1)*zeta(s)/zeta(4s). - _R. J. Mathar_, Apr 09 2011
%F A053164 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
%F A053164 a(n) = Sum_{d^4|n} phi(d). - _Ridouane Oudra_, Dec 31 2020
%F A053164 G.f.: Sum_{k>=1} phi(k) * x^(k^4) / (1 - x^(k^4)). - _Ilya Gutkovskiy_, Aug 20 2021
%e A053164 a(32) = 2 since 2 = 16^(1/4) and 16 is the largest 4th power dividing 32.
%p A053164 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
%t A053164 f[list_] := list[[1]]^Quotient[list[[2]], 4]; Table[Apply[Times, Map[f,FactorInteger[n]]], {n, 1, 81}] (* _Geoffrey Critzer_, Jan 21 2015 *)
%o A053164 (Scheme, with memoization macro definec)
%o A053164 (definec (A053164 n) (if (= 1 n) n (* (expt (A020639 n) (A002265 (A067029 n))) (A053164 (A028234 n)))))
%o A053164 (define (A002265 n) (floor->exact (/ n 4))) ;; For MIT/GNU Scheme
%o A053164 ;; _Antti Karttunen_, Sep 13 2017
%Y A053164 Cf. A000188, A000190, A008835, A053150.
%K A053164 nonn,mult
%O A053164 1,16
%A A053164 _Henry Bottomley_, Feb 29 2000
%E A053164 More terms from _Antti Karttunen_, Sep 13 2017