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Smallest 4th power divisible by n.
12

%I #23 Oct 27 2022 07:31:27

%S 1,16,81,16,625,1296,2401,16,81,10000,14641,1296,28561,38416,50625,16,

%T 83521,1296,130321,10000,194481,234256,279841,1296,625,456976,81,

%U 38416,707281,810000,923521,256,1185921,1336336,1500625,1296,1874161,2085136

%N Smallest 4th power divisible by n.

%H Amiram Eldar, <a href="/A053167/b053167.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) = (n/A000190(n))^4 = (n*A007913(n))^2/A008835(n*A007913(n)).

%F From _Amiram Eldar_, Jul 29 2022: (Start)

%F Multiplicative with a(p^e) = p^(e + ((4-e) mod 4)).

%F Sum_{n>=1} 1/a(n) = Product_{p prime} ((p^4+3)/(p^4-1)) = 1.341459051107600424... . (End)

%F 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 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 (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 Cf. A000190, A007913, A008835.

%Y Cf. A013662, A013674.

%K nonn,easy,mult

%O 1,2

%A _Henry Bottomley_, Feb 29 2000