A197863 -id:A197863 - cp's OEIS frontend

A197881 Number of times n occurs in A197863.

1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Franklin T. Adams-Watters, Oct 18 2011

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A197863, A112526.

f[p_, e_] := Switch[e, 1, 0, 2, 2, , 1]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 28 2023 *)

a(n)=my(es=factor(n)[,2]~);prod(k=1,#es,if(es[k]==1,0,if(es[k]==2,2,1)))

(define (A197881 n) (cond ((= 1 n) n) (else (* (case (A067029 n) ((1) 0) ((2) 2) (else 1)) (A197881 (A028234 n)))))) ;; Antti Karttunen, Jul 23 2017

Multiplicative with a(p^e) = 0 if e = 1, 2 if e = 2, and 1 otherwise.

More terms from Antti Karttunen, Jul 23 2017

A055231 Powerfree part of n: product of primes that divide n only once.

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 1, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 3, 1, 26, 1, 7, 29, 30, 31, 1, 33, 34, 35, 1, 37, 38, 39, 5, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 2, 55, 7, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71, 1, 73, 74, 3, 19, 77, 78, 79, 5

Offset: 1

Labos Elemer, Jun 21 2000

The previous name was: Write n = K^2*F where F is squarefree and F = g*f where g = gcd(K,F) and f = F/g; then a(n) = f(n) = F(n)/g(n). Thus gcd(K^2,f) = 1.
Differs from A007913; they coincide if and only if g(n) = 1.
a(n) is the powerfree part of n; i.e., if n=Product(pi^ei) over all i (prime factorization) then a(n)=Product(pi^ei) over those i with ei=1; if n=b*c^2*d^3 then a(n) is minimum possible value of b. - Henry Bottomley, Sep 01 2000
Also denominator of n/rad(n)^2, where rad is the squarefree kernel of n (A007947), numerator: A062378. - Reinhard Zumkeller, Dec 10 2002
Largest unitary squarefree number dividing n (the unitary squarefree kernel of n). - Steven Finch, Mar 01 2004
From Bernard Schott, Dec 19 2022: (Start)
a(n) = 1 iff n is a squareful number (A001694).
1 < a(n) < n iff n is a nonsquarefree number that is not squareful (A332785).
a(n) = n iff n is a squarefree number (A005117). (End)

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Positions of 1's: A001694.
Cf. A008833, A007913, A007947, A000188, A057521, A055773 (computed for n!), A056169 (number of prime divisors), A056671 (number of divisors), A092261 (sum of divisors of the n-th term), A197863, A332785.
Cf. A005117 (subsequence).

A055231 := proc(n)
    a := 1 ;
    if n > 1 then
        for f in ifactors(n)[2] do
            if op(2, f) = 1 then
                a := a*op(1, f) ;
            end if;
        end do:
    end if;
    a ;
end proc: # R. J. Mathar, Dec 23 2011

rad[n_] := Times @@ First /@ FactorInteger[n]; a[n_] := Denominator[n/rad[n]^2]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 20 2013, after Reinhard Zumkeller *)
f[p_, e_] := If[e==1, p, 1]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)

A055231(n)={
   local(a=1);
   f=factor(n) ;
   for(i=1,matsize(f)[1],
         if( f[i,2] ==1, a *=  f[i,1]
         )
   ) ;
   a ;
} /* R. J. Mathar, Mar 12 2012 */

a(n) = {my(f=factor(n)); for (k=1, #f~, if (f[k,2] > 1, f[k,2] = 0);); factorback(f);} \\ Michel Marcus, Aug 27 2017

from math import prod
from sympy import factorint
def A055231(n): return prod(p for p, e in factorint(n).items() if e == 1) # Chai Wah Wu, Nov 14 2022

;; With memoization-macro definec.
(definec (A055231 n) (if (= 1 n) 1 (* (if (= 1 (A067029 n)) (A020639 n) 1) (A055231 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = A007913(n)/gcd(A008833(n), A007913(n)).
a(n) = n/A057521(n).
Multiplicative with a(p) = p and a(p^e) = 1 for e > 1. - Vladeta Jovovic, Nov 01 2001
Dirichlet g.f.: zeta(s)*Product_{primes p} (1 + p^(1-s) - p^(-s) - p^(1-2s) + p^(-2s)). - R. J. Mathar, Dec 21 2011
a(n) = A007947(n)/A071773(n). - observed by Velin Yanev, Aug 27 2017, confirmed by Antti Karttunen, Nov 28 2017
a(1) = 1; for n > 1, a(n) = A020639(n)^A063524(A067029(n)) * a(A028234(n)). - Antti Karttunen, Nov 28 2017
a(n*m) = a(n)*a(m)/(gcd(n,a(m))*gcd(m,a(n))) for all n and m > 0 (conjectured). - Velin Yanev, Feb 06 2019. [This follows easily from the comment of Vladeta Jovovic. - N. J. A. Sloane, Mar 14 2019]
From Vaclav Kotesovec, Dec 19 2019: (Start)
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 - p^(1-3*s) + p^(2-3*s) - p^(2-2*s) + p^(-2*s) - p^(-s)).
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = Product_{primes p} (1 - 2/p^2 + 2/p^4 - 1/p^5) = 0.394913518073109872954607634745304266741971541072... (End)
a(n) = A197863(n)/n. - Amiram Eldar, Sep 01 2023

Name replaced with a simpler description (based on Henry Bottomley's comment) by Antti Karttunen, Nov 28 2017

Incorrect comments and example deleted by Peter Munn, Nov 30 2022

A007948 Largest cubefree number dividing n.

Original entry on oeis.org

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

A356191 a(n) is the smallest exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 32, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 32, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 96, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65

Offset: 1

Amiram Eldar, Jul 29 2022

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

Cf. A064549, A007913, A268335, A336643, A350390.

Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356192, A356193, A356194.

f[p_, e_] := If[OddQ[e], p^e, 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]%2, f[i,1]^f[i,2], f[i,1]^(f[i,2]+1)))};

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023

Multiplicative with a(p^e) = p^e if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A268335.
a(n) = A064549(n)/A007913(n).
a(n) = n*A336643(n).
a(n) = n^2/A350390(n).
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - p^(6-5*s) + p^(7-5*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(3-3*s) - p^(4-3*s) - 2*p^(2-2*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(2) * n^2 / 24 * (log(n) + 3*gamma - 1/2 + 12*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,
f'(2) = f(2) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.7165968208567630786609552448708126340725121316268495170070986645608062483...
and gamma is the Euler-Mascheroni constant A001620. (End)

A356192 a(n) is the smallest cubefull exponentially odd number (A335988) that is divisible by n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 32, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336

Offset: 1

Amiram Eldar, Jul 29 2022

First differs from A053149 and A356193 at n=16.

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

Cf. A001694, A013664, A268335, A335988.

Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356191, A356193, A356194.

f[p_, e_] := If[OddQ[e], p^Max[e, 3], p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2]%2, f[i,1]^max(f[i,2],3), f[i,1]^(f[i,2]+1)))};

Multiplicative with a(p^e) = p^max(e,3) if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A335988.
a(n) = A356191(n) iff n is a powerful number (A001694).
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p^2-1)/(p^3*(p^2-1))) = 1.69824776889117043774... .
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(6)/4) * Product_{p prime} (1 - 1/p^2 + 1/p^5 - 2/p^6 + 1/p^8 + 1/p^9 - 1/p^10) = 0.1559368144... . - Amiram Eldar, Nov 13 2022

A087320 Smallest perfect power (at least a square) that is a multiple of n.

Original entry on oeis.org

1, 4, 9, 4, 25, 36, 49, 8, 9, 100, 121, 36, 169, 196, 225, 16, 289, 36, 361, 100, 441, 484, 529, 144, 25, 676, 27, 196, 841, 900, 961, 32, 1089, 1156, 1225, 36, 1369, 1444, 1521, 400, 1681, 1764, 1849, 484, 225, 2116, 2209, 144, 49, 100, 2601, 676, 2809, 216, 3025

Offset: 1

Amarnath Murthy, Sep 03 2003

The prime signature of a(n) is not determined by the prime signature of n. For example, a(2^3*3^5) = 2^3*3^6, but a(2^3*5^5) = 2^5*5^5. - David Wasserman, May 03 2005

Likewise, this sequence is not multiplicative. The smallest exception is a(24) = 144 > 72 = a(3)*a(8). - Franklin T. Adams-Watters, Oct 18 2011

			a(54) = 216 = 6^3. 54 is the least n such that a(n)/n does not divide A007947(n).

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

Cf. A001597, A007947, A087321, A160400, A197863.

a[n_] := n * SelectFirst[Range[n], GCD @@ FactorInteger[n*#][[;; , 2]] > 1 &]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

a(n)/n <= A007947(n) (the squarefree kernel of n).
a(p^k) = p^k, a(k) = A007947(k)^2 for cubefree k. Furthermore, the upper bound on a(n)/n can be tightened to A007913(n). - Charlie Neder, Dec 26 2018
From Amiram Eldar, Jul 09 2022: (Start)
a(n) = n iff n is in A001597.
a(n) = n * A160400(n). (End)

Edited and extended by David Wasserman, May 03 2005

A356193 a(n) is the smallest cubefull number (A036966) that is a multiple of n.

Original entry on oeis.org

1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 16, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 32, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648, 3375, 97336

Offset: 1

Amiram Eldar, Jul 29 2022

First differs from A053149 and A356192 at n=16.

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

Cf. A002117, A036966, A360541.

Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356194.

f[p_, e_] := p^Max[e, 3]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]

a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^max(f[i,2],3))};

Multiplicative with a(p^e) = p^max(e,3).
a(n) = n iff n is in A036966.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + (3*p-2)/(p^3*(p-1))) = 1.76434793373691907811... . - Amiram Eldar, Jul 29 2022
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(3)/4) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^5 - 1/p^6 - 1/p^8 + 2/p^9 - 1/p^10) = 0.1559111567... . - Amiram Eldar, Nov 13 2022
a(n) = n * A360541(n). - Amiram Eldar, Sep 01 2023

A356194 a(n) is the smallest multiple of n whose prime factorization exponents are all powers of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 48, 25, 26, 81, 28, 29, 30, 31, 256, 33, 34, 35, 36, 37, 38, 39, 80, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 162, 55, 112, 57, 58, 59, 60, 61, 62, 63, 256, 65, 66, 67

Offset: 1

Amiram Eldar, Jul 29 2022

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

Cf. A138302, A353897.

Similar sequences: A053143, A053149, A053167, A066638, A087320, A087321, A197863, A356191, A356192, A356193.

f[p_, e_] := p^(2^Ceiling[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = {my(e=logint(n,2)); if(n == 2^e, n, 2^(e+1))};
a(n) = {my(f=factor(n)); prod(i=1, #f~, f[i,1]^s(f[i,2]))};

Multiplicative with a(p^e) = p^(2^ceiling(log_2(e))).

a(n) = n iff n is in A138302.

A369716 The number of divisors of the smallest powerful number that is a multiple of n.

Original entry on oeis.org

1, 3, 3, 3, 3, 9, 3, 4, 3, 9, 3, 9, 3, 9, 9, 5, 3, 9, 3, 9, 9, 9, 3, 12, 3, 9, 4, 9, 3, 27, 3, 6, 9, 9, 9, 9, 3, 9, 9, 12, 3, 27, 3, 9, 9, 9, 3, 15, 3, 9, 9, 9, 3, 12, 9, 12, 9, 9, 3, 27, 3, 9, 9, 7, 9, 27, 3, 9, 9, 27, 3, 12, 3, 9, 9, 9, 9, 27, 3, 15, 5, 9, 3

Offset: 1

Amiram Eldar, Jan 30 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A001694, A197863, A369717, A369718, A369719.

f[p_, e_] := If[e == 1, 3, e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x == 1, 3, x+1), factor(n)[, 2]));

a(n) = A000005(A197863(n)).
Multiplicative with a(p) = 3 and a(p^e) = e+1 for e >= 2.
a(n) >= A000005(n), with equality if and only if n is powerful (A001694).
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 1/p^s - 2/p^(2*s) + 1/p^(3*s)).
From Vaclav Kotesovec, Jan 30 2024: (Start)
Dirichlet g.f.: zeta(s)^3 * Product_{p prime} (1 - 3/p^(2*s) + 3/p^(3*s) - 1/p^(4*s)).
Let f(s) = Product_{primes p} (1 - 3/p^(2*s) + 3/p^(3*s) - 1/p^(4*s)).
Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + log(n)*((3*gamma - 1)*f(1) + f'(1)) + f(1)*(1 - 3*gamma + 3*gamma^2 - 3*sg1) + (3*gamma - 1)*f'(1) + f''(1)/2), where
f(1) = Product_{primes p} (1 - 3/p^2 + 3/p^3 - 1/p^4) = 0.33718787379158997196169281615215824494915412775816393888028828465611936...,
f'(1) = f(1) * Sum_{primes p} (6*p^2 - 9*p + 4) * log(p) / (p^4 - 3*p^2 + 3*p - 1) = f(1) * 2.35603132119230949914708478515883136510141335620960622673206366...,
f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} (-p*(12*p^5 - 27*p^4 + 16*p^3 + 9*p^2 - 12*p + 3) * log(p)^2 / (p^4 - 3*p^2 + 3*p - 1)^2) = f'(1)^2/f(1) + f(1) * (-7.3049026768735124341194605967271037971153161932236518820258070165876...),
gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

A369717 The sum of divisors of the smallest powerful number that is a multiple of n.

Original entry on oeis.org

1, 7, 13, 7, 31, 91, 57, 15, 13, 217, 133, 91, 183, 399, 403, 31, 307, 91, 381, 217, 741, 931, 553, 195, 31, 1281, 40, 399, 871, 2821, 993, 63, 1729, 2149, 1767, 91, 1407, 2667, 2379, 465, 1723, 5187, 1893, 931, 403, 3871, 2257, 403, 57, 217, 3991, 1281, 2863

Offset: 1

Amiram Eldar, Jan 30 2024

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

Cf. A000203, A001694, A197863, A369716, A369718, A369720.

Cf. A002117, A013661, A183699.

f[p_, e_] := If[e == 1, p^2 + p + 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i,2] == 1, f[i,2] = 2)); sigma(f);}

a(n) = A000203(A197863(n)).
Multiplicative with a(p) = p^2 + p + 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 2.
a(n) >= A000203(n), with equality if and only if n is powerful (A001694).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(3*s-3)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^6 - 1/p^7) = 1.01304866467771286896... .

cp's OEIS Frontend

A197881 Number of times n occurs in A197863.

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A055231 Powerfree part of n: product of primes that divide n only once.

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A007948 Largest cubefree number dividing n.

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A356191 a(n) is the smallest exponentially odd number that is divisible by n.

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A356192 a(n) is the smallest cubefull exponentially odd number (A335988) that is divisible by n.

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A087320 Smallest perfect power (at least a square) that is a multiple of n.

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A356193 a(n) is the smallest cubefull number (A036966) that is a multiple of n.

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