A058035 -id:A058035 - cp's OEIS frontend

A007948 Largest cubefree number dividing n.

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

Offset: 1

R. Muller

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative functions.
Florentin Smarandache, Only Problems, Not Solutions!, 1993.

Cf. A003557, A004709, A007947, A027748, A058035, A062378, A065465, A124010, A197863.

a007948 = last . filter ((== 1) . a212793) . a027750_row
-- Reinhard Zumkeller, May 27 2012, Jan 06 2012

Table[Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> p^Min[e, 2]], {n, 73}] (* Michael De Vlieger, Jul 18 2017 *)

a(n) = my(f=factor(n)); for (i=1, #f~, f[i,2] = min(f[i, 2], 2)); factorback(f); \\ Michel Marcus, Jun 09 2014
(Scheme, with memoization-macro definec) (definec (A007948 n) (if (= 1 n) n (* (expt (A020639 n) (min 2 (A067029 n))) (A007948 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

Multiplicative with a(p^e) = p^(min(e, 2)). - David W. Wilson, Aug 01 2001
a(n) = max{A212793(A027750(n,k)) * A027750(n,k): k=1..A000005(n)}. - Reinhard Zumkeller, May 27 2012
a(n) = A071773(n)*A007947(n). - observed by Velin Yanev, Aug 20 2017, confirmed by Antti Karttunen, Nov 28 2017
a(n) = n / A062378(n) = n / A003557(A003557(n)). - Antti Karttunen, Nov 28 2017
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465). - Amiram Eldar, Oct 13 2022

More terms from Henry Bottomley, Jun 18 2001

A252505 Number of biquadratefree (4th power free) divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 8, 2, 6, 6, 9

Offset: 1

Geoffrey Critzer, Mar 21 2015

Equivalently, a(n) is the number of divisors of n that are in A046100.
a(n) is also the number of divisors d such that the greatest common square divisor of d and n/d is 1.
The number of divisors d of n such that gcd(d, n/d) is squarefree. - Amiram Eldar, Aug 25 2023

			a(16) = 4 because there are 4 divisors of 16 that are 4th power free: 1,2,4,8.
a(16) = 4 because there are 4 divisors d of 16 such that the greatest common square divisor of d and 16/d is 1: 1,2,8,16.

Paul J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 37, Exercise 1.27.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A252505, Sequence Machine.
Eric Weisstein's World of Mathematics, Biquadratefree.

Cf. A046100 (biquadratefree numbers).
Cf. A034444 (squarefree divisors), A073184 (cubefree divisors).
Cf. A001620.
Cf. A000005, A058035, A307430.
Also obtained as a Dirichlet convolution of the following pairs: A034444 and A227291, A007427 and A286779, A008966 and A323308, A048691 and A363552, A271102 and A322327, A307445 and A370296, and A018892 and A378214 (conjectured).

Prepend[Table[Apply[Times, (FactorInteger[n][[All, 2]] /. x_ /; x > 3 -> 3) + 1], {n, 2, 100}], 1]

isA046100(n) = (n==1) || vecmax(factor(n)[, 2])<4;
a(n) = {d = divisors(n); sum(i=1, #d, isA046100(d[i]));} \\ Michel Marcus, Mar 22 2015

a(n) = vecprod(apply(x->min(x, 3) + 1, factor(n)[, 2])); \\ Amiram Eldar, Aug 25 2023

Dirichlet g.f.: zeta(s)^2/zeta(4*s).
Sum_{k=1..n} a(k) ~ 90*n/Pi^4 * (log(n) - 1 + 2*gamma - 360*zeta'(4)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
a(n) = Sum_{d|n} mu(gcd(d, n/d))^2. - Ilya Gutkovskiy, Feb 21 2020
Multiplicative with a(p^e) = min(e, 3) + 1. - Amiram Eldar, Sep 19 2020
From Antti Karttunen, May 14 2025: (Start)
Following formulas have been generated for this sequence by Sequence Machine:
a(n) = A000005(A058035(n)).
a(n) = Sum_{d|n} A307430(d).
a(n) = Sum_{d|n} A034444(d)*A227291(n/d).
a(n) = Sum_{d|n} A007427(d)*A286779(n/d).
a(n) = Sum_{d|n} A008966(d)*A323308(n/d).
a(n) = Sum_{d|n} A048691(d)*A363552(n/d).
a(n) = Sum_{d|n} A271102(d)*A322327(n/d).
a(n) = Sum_{d|n} A307445(d)*A370296(n/d).
a(n) = Sum_{d|n} A018892(d)*A378214(n/d). [Conjectured]
(End)

A365683 The largest exponentially squarefree divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Amiram Eldar, Sep 15 2023

First differs from A058035 at n = 32.

The number of these divisors is A365680(n) and their sum is A365682(n).

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

Cf. A070321, A209061, A365680, A365682.

f[p_, e_] := Module[{k = e}, While[! SquareFreeQ[k], k--]; p^k]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = {my(k = e); while(!issquarefree(k), k--); k;};
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}

Multiplicative with a(p^e) = p^A070321(e).
a(n) <= n, with equality if and only if n is exponentially squarefree number (A209061).
Sum_{k=1..n} a(k) ~ c*n^2, where c = 0.487850776747... = (1/2) * Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = A070321(k) and f(0) = 0.

A385006 The sum of the biquadratefree divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 15, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 15, 84, 144

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A365682 and A366992 at n = 32.

The number of these divisors is A252505(n), and the largest of them is A058035(n).

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

Cf. A046100, A048250, A058035, A073185, A252505, A365682, A366992.

The sum of divisors d of n such that d is: A000593 (odd), A033634 (exponentially odd), A035316 (square), A038712 (power of 2), A048250 (squarefree), A072079 (3-smooth), A073185 (cubefree), A113061 (cube), A162296 (nonsquarefree), A183097 (powerful), A186099 (5-rough), A353900 (exponentially 2^n), A385005 (cubefull), this sequence (biquadratefree).

f[p_, e_] := (p^Min[e+1, 4] - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; (p^min(e+1, 4) - 1)/(p - 1));}

Multiplicative with a(p^e) = (p^min(e+1, 4) - 1)/(p - 1).
In general, the sum of the k-free (numbers that are not divisible by a k-th power larger than 1) divisors of n is multiplicative with a(p^e) = (p^min(e+1, k) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(s-1) /zeta(4*s-4).
In general, the sum of the k-free divisors of n has Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(k*s-k).
Sum_{k=1..n} a(k) ~ (15/(2*Pi^2)) * n^2.
In general, the sum of the k-free divisors of n has an average order (Pi^2/(12*zeta(k))) * n^2.

A062379 n divided by largest 4th-power-free factor of 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, 4, 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, 8, 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, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Henry Bottomley, Jun 18 2001

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

Cf. A000190, A000583, A008835, A053164, A053165, A053166, A053167, A056553, A056554, A056555, A058035. See A003557 for squares and A062378 for cubes.

f[p_, e_] := p^Max[e-3, 0]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

(define (A062379 n) (/ n (A058035 n)))
(define (A058035 n) (if (= 1 n) n (* (expt (A020639 n) (min 3 (A067029 n))) (A058035 (A028234 n)))))
;; Antti Karttunen, Sep 13 2017

a(n) = n/A058035(n).

Multiplicative with a(p^e) = p^max(e-3, 0). - Amiram Eldar, Sep 07 2020

A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

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, 1, 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, 1, 65, 66, 67, 68, 69

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A053165 at n = 32 = 2^5: a(32) = 1 while A053165(32) = 2.
First differs from A383764 at n = 32 = 2^5: a(32) = 1 while A383764(32) = 32.
Equivalently, a(n) is the least divisor d of n such that n/d is a 4-full number (A036967).

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

Cf. A013661, A036967, A046100, A053165, A058035, A077610, A383764.

The largest unitary divisor of n that is: A000265 (odd), A006519 (power of 2), A055231 (squarefree), A057521 (powerful), A065330 (5-rough), A065331 (3-smooth), A350388 (square), A350389 (exponentially odd), A360539 (cubefree), A360540 (cubefull), A366126 (cube), A367168 (exponentially 2^n), this sequence (biquadratefree).

f[p_, e_] := If[e < 4, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] < 4, f[i, 1]^f[i, 2], 1)); }

a(n) = 1 if and only if n is a 4-full number (A036967).
a(n) = n if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = p^e if e <= 3, and 1 otherwise.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(-s) + p^(2-2*s) - p^(1-2*s) - p^(2-3*s) + p^(3-3*s) - p^(3-4*s) + p^(-4*s)).
Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^6 + 1/p^8 - 1/p^9) = 0.56331392082909224894... .

cp's OEIS Frontend

A007948 Largest cubefree number dividing n.

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A252505 Number of biquadratefree (4th power free) divisors of n.

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A365683 The largest exponentially squarefree divisor of n.

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A385006 The sum of the biquadratefree divisors of n.

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A062379 n divided by largest 4th-power-free factor of n.

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A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

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