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%I #43 Oct 27 2022 10:14:03
%S 1,2,3,2,5,6,7,2,3,10,11,6,13,14,15,2,17,6,19,10,21,22,23,6,5,26,3,14,
%T 29,30,31,4,33,34,35,6,37,38,39,10,41,42,43,22,15,46,47,6,7,10,51,26,
%U 53,6,55,14,57,58,59,30,61,62,21,4,65,66,67,34,69,70,71,6,73,74,15,38
%N Smallest positive integer for which n divides a(n)^4.
%C According to Broughan (2002, 2003, 2006), a(n) is the "upper 4th root of n". The "lower 4th root of n" is sequence A053164. - _Petros Hadjicostas_, Sep 15 2019
%H Amiram Eldar, <a href="/A053166/b053166.txt">Table of n, a(n) for n = 1..10000</a>
%H Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>.
%H Kevin A. Broughan, <a href="https://doi.org/10.4064/aa101-2-2">Restricted divisor sums</a>, Acta Arithmetica, 101(2) (2002), 105-114.
%H Kevin A. Broughan, <a href="http://ijpam.eu/contents/2003-5-3/2/2.pdf">Relationship between the integer conductor and k-th root functions</a>, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
%H Kevin A. Broughan, <a href="https://www.thebookshelf.auckland.ac.nz/document.php?wid=2937">Relaxations of the ABC Conjecture using integer k'th roots</a>, New Zealand J. Math. 35(2) (2006), 121-136.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmarandacheCeilFunction.html">Smarandache Ceil Function</a>.
%F a(n) = n/A000190(n) = A019554(n)/(A008835(A019554(n)^2))^(1/4).
%F If n is 5th-power-free (i.e., not 32, 64, 128, 243, ...) then a(n) = A007947(n).
%F Multiplicative with a(p^e) = p^(ceiling(e/4)). - _Christian G. Bower_, May 16 2005
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(7)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6) = 0.3528057925... . - _Amiram Eldar_, Oct 27 2022
%t f[p_, e_] := p^Ceiling[e/4]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 08 2020 *)
%o (PARI) a(n) = my(f=factor(n)); for (i=1, #f~, f[i,2] = ceil(f[i,2]/4)); factorback(f); \\ _Michel Marcus_, Jun 09 2014
%Y Cf. A000188 (inner square root), A019554 (outer square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A015052 (outer 5th root), A015053 (outer 6th root).
%Y Cf. A000189, A000190, A008835, A013665, A015051.
%K nonn,mult
%O 1,2
%A _Henry Bottomley_, Feb 29 2000
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