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A056553 Smallest 4th-power divisible by n divided by largest 4th-power which divides n.

%I A056553 #16 Oct 27 2022 07:31:56
%S A056553 1,16,81,16,625,1296,2401,16,81,10000,14641,1296,28561,38416,50625,1,
%T A056553 83521,1296,130321,10000,194481,234256,279841,1296,625,456976,81,
%U A056553 38416,707281,810000,923521,16,1185921,1336336,1500625,1296,1874161,2085136
%N A056553 Smallest 4th-power divisible by n divided by largest 4th-power which divides n.
%H A056553 Amiram Eldar, <a href="/A056553/b056553.txt">Table of n, a(n) for n = 1..10000</a>
%H A056553 Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>
%F A056553 a(n) = A053167(n)/A008835(n) = A056556(n)*A053165(n) = A056554(n)^4.
%F A056553 From _Amiram Eldar_, Oct 27 2022: (Start)
%F A056553 Multiplicative with a(p^e) = 1 if e is divisible by 4, and a(p^e) = p^4 otherwise.
%F A056553 Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(20)/(5*zeta(4))) * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^7 + 1/p^8) = 0.123026157003... . (End)
%e A056553 a(64) = 16 because smallest 4th power divisible by 64 is 256 and largest 4th power which divides 64 is 16 and 256/16 = 16.
%t A056553 f[p_, e_] := p^If[Divisible[e, 4], 0, 1]; a[n_] := (Times @@ (f @@@ FactorInteger[ n]))^4; Array[a, 100] (* _Amiram Eldar_, Aug 29 2019*)
%o A056553 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%4, f[i,1], 1))^4; } \\ _Amiram Eldar_, Oct 27 2022
%Y A056553 Cf. A000190, A008835, A053164, A053165, A053166, A053167, A056551, A056554, A056555.
%Y A056553 Cf. A013662, A013678.
%K A056553 nonn,mult
%O A056553 1,2
%A A056553 _Henry Bottomley_, Jun 25 2000