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Largest 4th power dividing n.
21

%I #41 Sep 01 2024 09:36:22

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,16,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,16,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,16,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,16,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,1,1,16,81

%N Largest 4th power dividing n.

%H Michael De Vlieger, <a href="/A008835/b008835.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) = A000188(A000188(n))^4.

%F Multiplicative with a(p^e) = p^(4[e/4]). - _Mitch Harris_, Apr 19 2005

%F Dirichlet g.f.: zeta(s) * zeta(4s-4) / zeta(4s). - _Álvar Ibeas_, Feb 12 2015

%F Sum_{k=1..n} a(k) ~ zeta(5/4) * n^(5/4) / (5*zeta(5)) - 45*n/Pi^4. - _Vaclav Kotesovec_, Feb 03 2019

%F a(n) = n/A053165(n). - _Amiram Eldar_, Aug 15 2023

%F a(n) = A053164(n)^4. - _Amiram Eldar_, Sep 01 2024

%p with(numtheory): [ seq( expand(nthpow(i,4)),i=1..200) ];

%t Max@ Select[Divisors@ #, IntegerQ@ Power[#, 1/4] &] & /@ Range@ 81 (* _Michael De Vlieger_, Mar 18 2015 *)

%t f[p_, e_] := p^(e - Mod[e, 4]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 15 2023 *)

%o (PARI) a(n) = {f = factor(n); for (i=1, #f~, f[i,2] = 4*(f[i,2]\4);); factorback(f);} \\ _Michel Marcus_, Mar 16 2015

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A008835(n): return prod(p**(e&-4) for p, e in factorint(n).items()) # _Chai Wah Wu_, Aug 08 2024

%Y Cf. A000188, A000190, A008833, A008834, A053164, A053165.

%K nonn,easy,mult

%O 1,16

%A _N. J. A. Sloane_

%E Entry improved by comments from _Henry Bottomley_, Feb 29 2000