A008835 -id:A008835 - cp's OEIS frontend

A336592 Numbers k such that k/A008835(k) is cubefree, where A008835(k) is the largest 4th power dividing k.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

Numbers such that none of the exponents in their prime factorization is of the form 4*m + 3.

Cohen (1963) proved that for a given number k > 2 the asymptotic density of numbers whose exponents in their prime factorization are not of the forms k*m - 1 is zeta(k)/zeta(k-1). In this sequence k = 4, and therefore its asymptotic density is zeta(4)/zeta(3) = Pi^4/(90*zeta(3)) = 0.9003926776...

			6 is a term since 6 = 2^1 * 3^1 and 1 is not of the form 4*m + 3.
8 is not a term since 8 = 2^3 and 3 is of the form 4*m + 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.

Cf. A002117, A004709, A008835, A013662.

Select[Range[100], Max[Mod[FactorInteger[#][[;; , 2]], 4]] < 3 &]

A336593 Numbers k such that k/A008835(k) is cubeful (A036966), where A008835(k) is the largest 4th power dividing k.

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 72, 88, 104, 108, 120, 125, 128, 135, 136, 152, 168, 184, 189, 200, 216, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 360, 375, 376, 378, 384, 392, 408, 424, 432, 440, 456, 459, 472, 488, 500, 504, 513, 520, 536, 540

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

Numbers such that at least one of the exponents in their prime factorization is of the form 4*m + 3.
The asymptotic density of this sequence is 1 - zeta(4)/zeta(3) = 0.0996073223... (Cohen, 1963).
The number of divisors of all the terms is divisible by 4.

			8 is a term since 8 = 2^3 and 3 is of the form 4*m + 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.

Complement of A336592.
Complement of A336594 within A252849.
Cf. A002117, A008835, A013662, A036966.
A176297 is a subsequence.

Select[Range[540], Max[Mod[FactorInteger[#][[;; , 2]], 4]] == 3 &]

A336594 Numbers k such that k/A008835(k) is cubefree but not squarefree (A067259), where A008835(k) is the largest 4th power dividing k.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 64, 68, 75, 76, 84, 90, 92, 98, 99, 100, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 192, 196, 198, 204, 207, 212, 220, 225, 228, 234, 236, 242, 244

Offset: 1

Amiram Eldar, Jul 26 2020

nonn

Numbers such that at least one of the exponents in their prime factorization is of the form 4*m + 2, and none are of the form 4*m + 3.

The asymptotic density of this sequence is zeta(4) * (1/zeta(3) - 1/zeta(2)) = Pi^4/(90*zeta(3)) - Pi^2/15 = 0.2424190509... (Cohen, 1963).

			4 is a term since the largest 4th power dividing 4 is 1, and 4/1 = 4 = 2^2 is cubefree but not squarefree.
64 is a term since the largest 4th power dividing 64 is 16, and 64/16 = 4 = 2^2 is cubefree but not squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical Notes, XIII. A Sequal to Note IV, Elemente der Mathematik, Vol. 18 (1963), pp. 8-11.

Cf. A002117, A008835, A013661, A013662, A067259, A182358.
Complement of A336593 within A252849.
A030140 is a subsequence.

Select[Range[250], Max[Mod[FactorInteger[#][[;; , 2]], 4]] == 2 &]

A008833 Largest square dividing n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 36, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 16, 1

Offset: 1

N. J. A. Sloane

The Dirichlet generating function of the arithmetic function of the largest t-th power dividing n is zeta(s)*zeta(t*s-t)/zeta(s*t), here with t=2 and in A008834 and A008835 with t=3 and t=4, respectively. - R. J. Mathar, Feb 19 2011

Daniel Forgues, Table of n, a(n) for n = 1..100000
Henry Bottomley, Some Smarandache-type multiplicative sequences
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions arXiv:1106.4038 [math.NT], 2011-2012, Remark 16.
Andrew Reiter, On (mod n) spirals, 2014, see also posting to Number Theory Mailing List, Mar 23 2014.
Eric Weisstein's World of Mathematics, Square part

Cf. A000188, A005563, A007913, A019554, A059897, A068310.

Cf. A008836, A358272, A001615.

a008833 n = head $ filter ((== 0) . (mod n)) $
   reverse $ takeWhile (<= n) $ tail a000290_list
-- Reinhard Zumkeller, Nov 13 2011

A008833 := proc(n)
    expand(numtheory:-nthpow(n,2)) ;
end proc:
seq(A008833(n), n=1..100) ;

a[n_] := First[ Select[ Reverse[ Divisors[n]], IntegerQ[Sqrt[#]]&, 1]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 12 2011 *)
f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jul 07 2020 *)

A008833(n)=n/core(n) \\ Michael B. Porter, Oct 17 2009

from sympy.ntheory.factor_ import core
def A008833(n): return n//core(n) # Chai Wah Wu, Dec 30 2021

a(n) = A000188(n)^2 = n/A007913(n). Cf. A019554.
Multiplicative with a(p^e) = p^(2[e/2]). - David W. Wilson, Aug 01 2001
Dirichlet g.f.: zeta(s)*zeta(2s-2)/zeta(2s). - R. J. Mathar, Oct 31 2011
a(n) = A005563(n-1) / A068310(n) for n > 1. - Reinhard Zumkeller, Nov 26 2011
Sum_{k=1..n} a(k) ~ Zeta(3/2) * n^(3/2) / (3*Zeta(3)). - Vaclav Kotesovec, Feb 01 2019
a(A059897(n,k)) = A059897(a(n), a(k)). - Peter Munn, Nov 30 2019
From Ridouane Oudra, May 11 2025: (Start)
a(n) = Sum_{d|n} lambda(d)*d*psi(n/d), where lambda = A008836 and psi = A001615.
a(n) = lambda(n) * Sum_{d|n} lambda(d)*d*phi(n/d).
a(n) = A008836(n) * A358272(n). (End)

A053164 4th root of largest 4th power dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Henry Bottomley, Feb 29 2000

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

			a(32) = 2 since 2 = 16^(1/4) and 16 is the largest 4th power dividing 32.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.

Cf. A000188, A000190, A008835, A053150.

A053164 := proc(n) local a,f,e,p ; for f in ifactors(n)[2] do e:= op(2,f) ; p := op(1,f) ; a := a*p^floor(e/4) ; end do ; a ; end proc: # R. J. Mathar, Jan 11 2012

f[list_] := list[[1]]^Quotient[list[[2]], 4]; Table[Apply[Times, Map[f,FactorInteger[n]]], {n, 1, 81}] (* Geoffrey Critzer, Jan 21 2015 *)

a(n) = A000188(A000188(n)) = A008835(n)^(1/4).
Multiplicative with a(p^e) = p^[e/4].
Dirichlet g.f.: zeta(4s-1)*zeta(s)/zeta(4s). - R. J. Mathar, Apr 09 2011
Sum_{k=1..n} a(k) ~ 90*zeta(3)*n/Pi^4 + 3*zeta(1/2)*sqrt(n)/Pi^2. - Vaclav Kotesovec, Dec 01 2020
a(n) = Sum_{d^4|n} phi(d). - Ridouane Oudra, Dec 31 2020
G.f.: Sum_{k>=1} phi(k) * x^(k^4) / (1 - x^(k^4)). - Ilya Gutkovskiy, Aug 20 2021

More terms from Antti Karttunen, Sep 13 2017

A053150 Cube root of largest cube dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Henry Bottomley, Feb 28 2000

This can be thought as a "lower 3rd root" of a positive integer. Upper k-th roots were studied by Broughan (2002, 2003, 2006). The sequence of "upper 3rd root" of positive integers is given by A019555. - Petros Hadjicostas, Sep 15 2019

Antti Karttunen, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114.
Kevin A. Broughan, Relationship between the integer conductor and k-th root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121-136.
Vaclav Kotesovec, Graph - the asymptotic ratio.

Cf. A000188 (inner square root), A019554 (outer square root), A019555 (outer third root), A053164 (inner 4th root), A053166 (outer 4th root), A015052 (outer 5th root), A015053 (outer 6th root).

Cf. A000189, A000190, A008834, A008835, A015051, A061704.

f[list_] := list[[1]]^Quotient[list[[2]], 3]; Table[Apply[Times, Map[f,FactorInteger[n]]], {n, 1, 81}] (* Geoffrey Critzer, Jan 21 2015 *)
Table[SelectFirst[Reverse@ Divisors@ n, IntegerQ[#^(1/3)] &]^(1/3), {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f[p_, e_] := p^Floor[e/3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

A053150(n) = { my(f = factor(n), m = 1); for (k=1, #f~, m *= (f[k, 1]^(f[k, 2]\3)); ); m; } \\ Antti Karttunen, Jul 28 2017

a(n) = my(f = factor(n)); for (k=1, #f~, f[k,2] = f[k,2]\3); factorback(f); \\ Michel Marcus, Jul 28 2017

from math import prod
from sympy import factorint
def A053150(n): return prod(p**(q//3) for p, q in factorint(n).items()) # Chai Wah Wu, Aug 18 2021

Multiplicative with a(p^e) = p^[e/3]. - Mitch Harris, Apr 19 2005
a(n) = A008834(n)^(1/3) = sqrt(A000189(n)/A000188(A050985(n))).
Dirichlet g.f.: zeta(3s-1)*zeta(s)/zeta(3s). - R. J. Mathar, Apr 09 2011
Sum_{k=1..n} a(k) ~ Pi^2 * n / (6*zeta(3)) + 3*zeta(2/3) * n^(2/3) / Pi^2. - Vaclav Kotesovec, Jan 31 2019
a(n) = Sum_{d^3|n} phi(d). - Ridouane Oudra, Dec 30 2020
G.f.: Sum_{k>=1} phi(k) * x^(k^3) / (1 - x^(k^3)). - Ilya Gutkovskiy, Aug 20 2021

More terms from Antti Karttunen, Jul 28 2017

A053165 4th-power-free part of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 2, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Henry Bottomley, Feb 29 2000

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.

Equivalent sequences for other powers: A007913 (2), A050985 (3).
Cf. A000190, A000203, A008835.
A003961, A059897 are used to express relationship between terms of this sequence.
Related to A065331 via A225546.

f[p_, e_] := p^Mod[e, 4]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

a(n)=my(f=factor(n)); f[,2]=f[,2]%4; factorback(f) \\ Charles R Greathouse IV, Sep 02 2015

from operator import mul
from functools import reduce
from sympy import factorint
def A053165(n):
    return 1 if n <=1 else reduce(mul,[p**(e % 4) for p,e in factorint(n).items()])
# Chai Wah Wu, Feb 04 2015

a(n) = n/A008835(n).
Dirichlet g.f.: zeta(4s)*zeta(s-1)/zeta(4s-4). The Dirichlet convolution of this sequence with A008835 is A000203. - R. J. Mathar, Apr 05 2011
From Peter Munn, Jan 15 2020: (Start)
a(2) = 2; a(4) = 4; a(n^4) = 1; a(A003961(n)) = A003961(a(n)); a(A059897(n,k)) = A059897(a(n), a(k)).
a(A225546(n)) = A225546(A065331(n)).
(End)
Multiplicative with a(p^e) = p^(e mod 4). - Amiram Eldar, Sep 07 2020
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 210. - Vaclav Kotesovec, Aug 20 2021

A053166 Smallest positive integer for which n divides a(n)^4.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 4, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 4, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38

Offset: 1

Henry Bottomley, Feb 29 2000

According to Broughan (2002, 2003, 2006), a(n) is the "upper 4th root of n". The "lower 4th root of n" is sequence A053164. - Petros Hadjicostas, Sep 15 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114.
Kevin A. Broughan, Relationship between the integer conductor and k-th root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121-136.
Eric Weisstein's World of Mathematics, Smarandache Ceil Function.

Cf. A000188 (inner square root), A019554 (outer square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A015052 (outer 5th root), A015053 (outer 6th root).

Cf. A000189, A000190, A008835, A013665, A015051.

f[p_, e_] := p^Ceiling[e/4]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 08 2020 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,2] = ceil(f[i,2]/4)); factorback(f); \\ Michel Marcus, Jun 09 2014

a(n) = n/A000190(n) = A019554(n)/(A008835(A019554(n)^2))^(1/4).
If n is 5th-power-free (i.e., not 32, 64, 128, 243, ...) then a(n) = A007947(n).
Multiplicative with a(p^e) = p^(ceiling(e/4)). - Christian G. Bower, May 16 2005
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(7)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6) = 0.3528057925... . - Amiram Eldar, Oct 27 2022

A053167 Smallest 4th power divisible by n.

Original entry on oeis.org

1, 16, 81, 16, 625, 1296, 2401, 16, 81, 10000, 14641, 1296, 28561, 38416, 50625, 16, 83521, 1296, 130321, 10000, 194481, 234256, 279841, 1296, 625, 456976, 81, 38416, 707281, 810000, 923521, 256, 1185921, 1336336, 1500625, 1296, 1874161, 2085136

Offset: 1

Henry Bottomley, Feb 29 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.

Cf. A000190, A007913, A008835.

Cf. A013662, A013674.

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*)

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

a(n) = (n/A000190(n))^4 = (n*A007913(n))^2/A008835(n*A007913(n)).
From Amiram Eldar, Jul 29 2022: (Start)
Multiplicative with a(p^e) = p^(e + ((4-e) mod 4)).
Sum_{n>=1} 1/a(n) = Product_{p prime} ((p^4+3)/(p^4-1)) = 1.341459051107600424... . (End)
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

A337533 1 together with nonsquares whose square part's square root is in the sequence.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68

Offset: 1

Peter Munn, Aug 31 2020

The appearance of a number is determined by its prime signature.
Every squarefree number is present, as the square root of the square part of a squarefree number is 1. Other 4th-power-free numbers are present if and only if they are nonsquare.
If the square part of nonsquarefree k is a 4th power, k does not appear.
Every positive integer k is the product of a unique subset S_k of the terms of A050376, which are arranged in array form in A329050 (primes in column 0, squares of primes in column 1, 4th powers of primes in column 2 and so on). k > 1 is in this sequence if and only if the members of S_k occur in consecutive columns of A329050, starting with column 0.
If the qualifying condition in the previous paragraph was based on the rows instead of the columns of A329050, we would get A055932. The self-inverse function defined by A225546 transposes A329050. A225546 also has multiplicative properties such that if we consider A055932 and this sequence as sets, A225546(.) maps the members of either set 1:1 onto the other set.

			4 is square and not 1, so 4 is not in the sequence.
12 = 3 * 2^2 is nonsquare, and has square part 4, whose square root (2) is in the sequence. So 12 is in the sequence.
32 = 2 * 4^2 is nonsquare, but has square part 16, whose square root (4) is not in the sequence. So 32 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square part
Index to sequences related to prime signature

Complement of A337534.
Closed under A000188(.).
A209229, A267116 are used in a formula defining this sequence.
Subsequences: A005117, A036537, A179646, A179666, A179669, A189987, A191554, A191555, A298207, A317090.
Subsequence of A164514.
A007913, A008833, A008835, A335324 give the squarefree, square and comparably related parts of a number.
Cf. A050376, A329050.
Related to A055932 via A225546.

S:= {1}:
for n from 2 to 100 do
  if not issqr(n) then
    F:= ifactors(n)[2];
    s:= mul(t[1]^floor(t[2]/2),t=F);
    if member(s,S) then S:= S union {n} fi
  fi
od:
sort(convert(S,list)); # Robert Israel, Jan 07 2025

pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[100], # == 1 || pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* Amiram Eldar, Sep 18 2020 *)

Numbers m such that A209229(A267116(m) + 1) = 1.
If A008835(a(n)) > 1 then A335324(a(n)) > 1.
If A008833(a(n)) > 1 then A007913(a(n)) > 1.

cp's OEIS Frontend

A336592 Numbers k such that k/A008835(k) is cubefree, where A008835(k) is the largest 4th power dividing k.

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A336593 Numbers k such that k/A008835(k) is cubeful (A036966), where A008835(k) is the largest 4th power dividing k.

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A336594 Numbers k such that k/A008835(k) is cubefree but not squarefree (A067259), where A008835(k) is the largest 4th power dividing k.

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A008833 Largest square dividing n.

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A053164 4th root of largest 4th power dividing n.

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A053150 Cube root of largest cube dividing n.

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A053165 4th-power-free part of n.

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