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A053167 Smallest 4th power divisible by n.

%I A053167 #23 Oct 27 2022 07:31:27
%S A053167 1,16,81,16,625,1296,2401,16,81,10000,14641,1296,28561,38416,50625,16,
%T A053167 83521,1296,130321,10000,194481,234256,279841,1296,625,456976,81,
%U A053167 38416,707281,810000,923521,256,1185921,1336336,1500625,1296,1874161,2085136
%N A053167 Smallest 4th power divisible by n.
%H A053167 Amiram Eldar, <a href="/A053167/b053167.txt">Table of n, a(n) for n = 1..10000</a>
%H A053167 Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>.
%F A053167 a(n) = (n/A000190(n))^4 = (n*A007913(n))^2/A008835(n*A007913(n)).
%F A053167 From _Amiram Eldar_, Jul 29 2022: (Start)
%F A053167 Multiplicative with a(p^e) = p^(e + ((4-e) mod 4)).
%F A053167 Sum_{n>=1} 1/a(n) = Product_{p prime} ((p^4+3)/(p^4-1)) = 1.341459051107600424... . (End)
%F A053167 Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(16)/(5*zeta(4))) * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^7 + 1/p^8) = 0.1230279197... . - _Amiram Eldar_, Oct 27 2022
%t A053167 f[p_, e_] := p^(e + Mod[4 - Mod[e, 4], 4]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Aug 29 2019*)
%o A053167 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] + (4-f[i,2])%4));} \\ _Amiram Eldar_, Oct 27 2022
%Y A053167 Cf. A000190, A007913, A008835.
%Y A053167 Cf. A013662, A013674.
%K A053167 nonn,easy,mult
%O A053167 1,2
%A A053167 _Henry Bottomley_, Feb 29 2000