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Smallest number k (k>0) such that n*k is a perfect 4th power.
4

%I #20 Oct 27 2022 07:32:04

%S 1,8,27,4,125,216,343,2,9,1000,1331,108,2197,2744,3375,1,4913,72,6859,

%T 500,9261,10648,12167,54,25,17576,3,1372,24389,27000,29791,8,35937,

%U 39304,42875,36,50653,54872,59319,250,68921,74088,79507,5324,1125

%N Smallest number k (k>0) such that n*k is a perfect 4th power.

%H Amiram Eldar, <a href="/A056555/b056555.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>.

%F a(n) = A053167(n)/n = n^3/A000190(n)^4 = A056553(n)/A053165(n).

%F Multiplicative with a(p^e) = p^((4 - e) mod 4). - _Amiram Eldar_, Sep 08 2020

%F Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(16)/(4*zeta(4))) * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^7 + 1/p^8) = 0.1537848996... . - _Amiram Eldar_, Oct 27 2022

%e a(64) = 4 because the smallest 4th power divisible by 64 is 256 and 64*4 = 256.

%t f[p_, e_] := p^Mod[4 - e, 4]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 08 2020 *)

%o (PARI) a(n,f=factor(n))=f[,2]=-f[,2]%4; factorback(f) \\ _Charles R Greathouse IV_, Apr 24 2020

%Y Cf. A000190, A008835, A048798, A053164, A053165, A053166, A053167, A056554.

%Y Cf. A013662, A013674.

%K nonn,mult

%O 1,2

%A _Henry Bottomley_, Jun 25 2000