A008835 Largest 4th power dividing n.

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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 81

Offset

1,16

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

Formula

a(n) = A000188(A000188(n))^4.

Multiplicative with a(p^e) = p^(4[e/4]). - Mitch Harris, Apr 19 2005

Dirichlet g.f.: zeta(s) * zeta(4s-4) / zeta(4s). - Álvar Ibeas, Feb 12 2015

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

a(n) = n/A053165(n). - Amiram Eldar, Aug 15 2023

a(n) = A053164(n)^4. - Amiram Eldar, Sep 01 2024

Maple

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

Mathematica

Max@ Select[Divisors@ #, IntegerQ@ Power[#, 1/4] &] & /@ Range@ 81 (* Michael De Vlieger, Mar 18 2015 *)
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 *)

Prog

(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
(Python)
from math import prod
from sympy import factorint
def A008835(n): return prod(p**(e&-4) for p, e in factorint(n).items()) # Chai Wah Wu, Aug 08 2024

Crossrefs

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

Sequence in context: A265491 A209201 A050467 * A040259 A353806 A040260

Adjacent sequences: A008832 A008833 A008834 * A008836 A008837 A008838

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Extensions

Entry improved by comments from Henry Bottomley, Feb 29 2000

Status

approved