A008835 -id:A008835

A336593

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

+20
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

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.

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

Complement of A336592.

Complement of A336594 within A252849.

Cf. A002117, A008835, A013662, A036966.

A176297 is a subsequence.

KEYWORD

nonn

AUTHOR

Amiram Eldar, Jul 26 2020

STATUS

approved

A336594

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

+20
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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).

LINKS

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.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Cf. A002117, A008835, A013661, A013662, A067259, A182358.

Complement of A336593 within A252849.

A030140 is a subsequence.

KEYWORD

nonn

AUTHOR

Amiram Eldar, Jul 26 2020

STATUS

approved

A336592

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

+20
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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...

LINKS

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.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Cf. A002117, A004709, A008835, A013662.

KEYWORD

nonn

AUTHOR

Amiram Eldar, Jul 26 2020

STATUS

approved

A008833

Largest square dividing n.

+10
106

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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

LINKS

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

FORMULA

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

MAPLE

A008833 := proc(n)

expand(numtheory:-nthpow(n, 2)) ;

end proc:

seq(A008833(n), n=1..100) ;

MATHEMATICA

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 *)

PROG

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

(Haskell)

a008833 n = head $ filter ((== 0) . (mod n)) $

reverse $ takeWhile (<= n) $ tail a000290_list

-- Reinhard Zumkeller, Nov 13 2011

(Python)

from sympy.ntheory.factor_ import core

def A008833(n): return n//core(n) # Chai Wah Wu, Dec 30 2021

CROSSREFS

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

KEYWORD

nonn,easy,mult

AUTHOR

N. J. A. Sloane.

STATUS

approved

A053164

4th root of largest 4th power dividing n.

+10
23

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,16

COMMENTS

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

(Scheme, with memoization macro definec)

(definec (A053164 n) (if (= 1 n) n (* (expt (A020639 n) (A002265 (A067029 n))) (A053164 (A028234 n)))))

(define (A002265 n) (floor->exact (/ n 4))) ;; For MIT/GNU Scheme

;; Antti Karttunen, Sep 13 2017

CROSSREFS

Cf. A000188, A000190, A008835, A053150.

KEYWORD

nonn,mult

AUTHOR

Henry Bottomley, Feb 29 2000

EXTENSIONS

More terms from Antti Karttunen, Sep 13 2017

STATUS

approved

A053150

Cube root of largest cube dividing n.

+10
22

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

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

LINKS

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.

FORMULA

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

MATHEMATICA

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 *)

PROG

(PARI) 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

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

(Python 3.8+)

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

CROSSREFS

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.

KEYWORD

easy,nonn,mult

AUTHOR

Henry Bottomley, Feb 28 2000

EXTENSIONS

More terms from Antti Karttunen, Jul 28 2017

STATUS

approved

A053166

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

+10
15

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

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.

FORMULA

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

MATHEMATICA

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

PROG

(PARI) 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

CROSSREFS

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.

KEYWORD

nonn,mult

AUTHOR

Henry Bottomley, Feb 29 2000

STATUS

approved

A053165

4th-power-free part of n.

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

FORMULA

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

MATHEMATICA

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

PROG

(Python)

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

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

CROSSREFS

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.

KEYWORD

nonn,mult

AUTHOR

Henry Bottomley, Feb 29 2000

STATUS

approved

A053167

Smallest 4th power divisible by n.

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Henry Bottomley, Some Smarandache-type multiplicative sequences.

FORMULA

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

MATHEMATICA

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*)

PROG

(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

CROSSREFS

Cf. A000190, A007913, A008835.

Cf. A013662, A013674.

KEYWORD

nonn,easy,mult

AUTHOR

Henry Bottomley, Feb 29 2000

STATUS

approved

A335324

Square part of 4th-power-free part of n.

+10
6

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 1, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

Equivalently, biquadratefree (4th-power-free) part of square part of n.

Multiplicative. The terms are squares of squarefree numbers (A062503).

Every positive integer n is the product of a unique subset S_n of the terms of A050376 (sometimes called Fermi-Dirac primes). a(n) is the product of the members of S_n that are squares of prime numbers (A001248).

LINKS

Michel Marcus, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Biquadratefree.

Eric Weisstein's World of Mathematics, Square part.

FORMULA

a(n) = A053165(A008833(n)) = A008833(A053165(n)).

a(n) = A053165(n) / A007913(n).

a(n) = A008833(n) / A008835(n).

n = A007913(n) * a(n) * A008835(n).

a(n) = A225546(A038500(A225546(n))).

a(n^2) = A007913(n)^2.

a(A003961(n)) = A003961(a(n)).

a(A331590(n, k)) = A331590(a(n), a(k)).

a(p^e) = p^(2*floor(e/2) - 4*floor(e/4)). - Amiram Eldar, Jun 01 2020

From Amiram Eldar, Sep 21 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-2) * zeta(4*s)/(zeta(2*s) * zeta(4*s-4)).

Sum_{k=1..n} a(k) ~ (4*zeta(3/2)*zeta(4))/(21*zeta(3)) * n^(3/2). (End)

EXAMPLE

Removing the 4th powers from 192 = 2^6 * 3^1 gives 2^(6 - 4) * 3^1 = 2^2 * 3 = 12. So the 4th-power-free part of 192 is 12. The square part of 12 (largest square dividing 12) is 4. So a(192) = 4.

MATHEMATICA

f[p_, e_] := p^(2*Floor[e/2] - 4*Floor[e/4]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 01 2020 *)

PROG

(PARI) A053165(n)=my(f=factor(n)); f[, 2]=f[, 2]%4; factorback(f);

a(n) = my(m=A053165(n)); m/core(m); \\ Michel Marcus, Jun 01 2020

(Python)

from math import prod

from sympy import factorint

def A335324(n): return prod(p**(e&2) for p, e in factorint(n).items()) # Chai Wah Wu, Aug 07 2024

CROSSREFS

A007913, A008833, A008835, A053165 are used in formulas defining the sequence.

Column 1 of A352780.

Range of values is A062503.

Positions of 1's: A252895.

Related to A038500 by A225546.

The formula section details how the sequence maps the terms of A003961, A331590.

Cf. A001248, A050376, A083730.

Cf. A002117, A013662, A078434.

KEYWORD

nonn,easy,mult

AUTHOR

Peter Munn, May 31 2020

STATUS

approved