A057521 -id:A057521 - cp's OEIS frontend

A384048 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is squarefree.

1, 2, 3, 3, 5, 6, 7, 7, 8, 10, 11, 9, 13, 14, 15, 15, 17, 16, 19, 15, 21, 22, 23, 21, 24, 26, 26, 21, 29, 30, 31, 31, 33, 34, 35, 24, 37, 38, 39, 35, 41, 42, 43, 33, 40, 46, 47, 45, 48, 48, 51, 39, 53, 52, 55, 49, 57, 58, 59, 45, 61, 62, 56, 63, 65, 66, 67, 51

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A063659.
Cf. A000010, A005117, A055231, A057521, A065466, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), this sequence (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := If[e == 1, p, 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~, f[i,1]^f[i,2] - if(f[i,2] == 1, 0, 1));}

Multiplicative with a(p) = p and a(p^e) = p^e - 1 if e >= 2.
a(n) = n * A047994(n) / A384050(n).
a(n) = A047994(A057521(n)) * A055231(n) = A000010(A055231(n)) * A057521(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^3*(p+1))) = 0.947733... (A065466).

A384050 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a powerful number.

Original entry on oeis.org

1, 1, 2, 4, 4, 2, 6, 8, 9, 4, 10, 8, 12, 6, 8, 16, 16, 9, 18, 16, 12, 10, 22, 16, 25, 12, 27, 24, 28, 8, 30, 32, 20, 16, 24, 36, 36, 18, 24, 32, 40, 12, 42, 40, 36, 22, 46, 32, 49, 25, 32, 48, 52, 27, 40, 48, 36, 28, 58, 32, 60, 30, 54, 64, 48, 20, 66, 64, 44

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A384039.
Cf. A000010, A001694, A055231, A057521, A330596, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), this sequence (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[e < 2, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] == 1, 1, 0));}

Multiplicative with a(p) = p-1, and p^e if e >= 2.
a(n) = n * A047994(n) / A384048(n).
a(n) = A047994(A055231(n)) * A057521(n) = A000010(A055231(n)) * A057521(n).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(2*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).

A360540 a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 16, 81, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 11 2023

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

Cf. A004709, A036966, A057521, A360539.

f[p_, e_] := If[e > 2, p^e, 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], 1));}

a(n) = 1 if and only if n is a cubefree number (A004709).
a(n) = n if and only if n is a cubefull number (A036966).
a(n) <= A057521(n) with equality if and only if n is in A337050.
a(n) = n/A360539(n).
Multiplicative with a(p^e) = p^e if e >= 3, and 1 otherwise.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - p^(1-s) + p^(-s) - p^(1-3*s) - p^(1-2*s) + p^(-2*s) + p^(3-3*s)).

A195086 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.

Original entry on oeis.org

8, 24, 27, 36, 40, 54, 56, 88, 100, 104, 120, 125, 135, 136, 152, 168, 180, 184, 189, 196, 225, 232, 248, 250, 252, 264, 270, 280, 296, 297, 300, 312, 328, 343, 344, 351, 375, 376, 378, 396, 408, 424, 440, 441, 450, 456, 459, 468, 472, 484, 488

Offset: 1

Harvey P. Dale, Sep 08 2011

nonn

From Amiram Eldar, Nov 07 2020: (Start)
Numbers whose powerful part (A057521) is either a cube of a prime (A030078) or a square of a squarefree semiprime (A085986).
The asymptotic density of this sequence is (6/Pi^2) * (Sum_{p prime} 1/(p^2*(p+1)) + Sum_{p=4} (-1)^(k+1)*(k-1)*P(k) + (Sum_{k>=2} (-1)^k*P(k))^2)/2 = 0.0963023158..., where P is the prime zeta function. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Alfred Rényi, On the density of certain sequences of integers, Publications de l'Institut Mathématique, Vol. 8 (1955), pp. 157-162.
Index entries for sequences computed from exponents in factorization of n

Subsequence of A048108.
Subsequences: A030078 and A085986.
Cf. A001221, A001222, A025487, A057521, A060687, A195069, A195087, A195088, A195089, A195090, A195091, A195092, A195093, A046660, A257851, A261256, A264959.

a195086 n = a195086_list !! (n-1)
a195086_list = filter ((== 2) . a046660) [1..]
-- Reinhard Zumkeller, Nov 29 2015

Select[Range[500],PrimeOmega[#]-PrimeNu[#]==2&]

is(n)=bigomega(n)-omega(n)==2 \\ Charles R Greathouse IV, Sep 14 2015

is(n)=my(f=factor(n)[,2]); vecsum(f)==#f+2 \\ Charles R Greathouse IV, Aug 01 2016

A001222(a(n)) - A001221(a(n)) = 2.

A046660(a(n)) = 2. - Reinhard Zumkeller, Nov 29 2015

A071773 a(n) = gcd(rad(n), n/rad(n)), where rad(n) = A007947(n) is the squarefree kernel of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 7, 3, 10, 1, 1, 1, 2

Offset: 1

Reinhard Zumkeller, Jun 24 2002

n is squarefree iff a(n)=1.
Product of primes dividing n more than once. - Charles R Greathouse IV, Aug 08 2013
Squarefree kernel of the square part of n. - Peter Munn, Jun 12 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Eric Weisstein's World of Mathematics, Squarefree.

Cf. A003415, A003557, A005117, A007947, A007948, A008833, A057521, A166486 (parity of terms), A359433 (Dirichlet inverse).

Cf. A065464.

Table[With[{r = Apply[Times, FactorInteger[n][[All, 1]]]}, GCD[r, n/r]], {n, 104}] (* Michael De Vlieger, Aug 20 2017 *)

a(n)=my(f=factor(n));prod(i=1,#f~,f[i,1]^(f[i,2]>1)) \\ Charles R Greathouse IV, Aug 08 2013

;; With memoization-macro definec.
(definec (A071773 n) (if (= 1 n) n (* (if (zero? (modulo n (expt (A020639 n) 2))) (A020639 n) 1) (A071773 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = gcd(A007947(n), A003557(n)).
Multiplicative with p^e -> p^ceiling((e-1)/e), p prime.
a(n) = rad(n/rad(n)) = A007947(A003557(n)). - Velin Yanev, Antti Karttunen, Aug 20 2017, Nov 28 2017
a(n) = A007947(A057521(n)). - Antti Karttunen, Nov 28 2017
a(n) = A007947(A008833(n)). - Peter Munn, Jun 12 2020
a(n) = gcd(A003415(n), A007947(n)). - Antti Karttunen, Jan 02 2023
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(2*s-1) - 1/p^(2*s)). - Amiram Eldar, Nov 09 2023
From Vaclav Kotesovec, May 06 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(4*s-1) - 1/p^(4*s-2)).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A065464 = Product_{p prime} (1 - 2/p^2 + 1/p^3) = 0.428249505677094440218765707581823546121298513355936144031901379532123...
f'(1) = f(1) * Sum_{p prime} 2*(3*p-2)*log(p) / (p^3-2*p+1) = f(1) * 2.939073481649229666406787986900328729326669597518287791424059647447664...
and gamma is the Euler-Mascheroni constant A001620. (End)

A203025 Largest perfect power divisor of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 8, 25, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 27, 1, 8, 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

Offset: 1

Antonio Roldán, Dec 28 2011

nonn

This sequence shares many elements with A057521, but is not identical: A057521(72)=72 but a(72)=36.

Not multiplicative: a(49)=49; a(125)=125, a(49*125) = 1225 <> 49*125.

			a(40)=a(2^3*5)=2^3=8.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A057521, A087320, A274006.

A203025:=proc(n)
    local a,Le,d,i,k,pe;
    pe := ifactors(n)[2];
    Le := {seq(i[2],i=pe)} minus {1};
    a := 1;
    for k in Le do
        d := mul(i[1]^(k*floor(i[2]/k)), i=pe) ;
        a:=max(a,d);
    end do;
    a
end proc:
seq(A203025(n),n=1..10000); # Felix Huber, Jun 01 2025

Table[If[SquareFreeQ[n], 1, s = FactorInteger[n]; Max[Table[Times @@ Cases[s, {p_, ep_} :> p^i /; (ep >= i)], {i, 2, Max[s[[All, 2]]]}]]], {n, 100}] (* Olivier Gerard, Jun 03 2016 *)

a(n)=my(f=factor(n),mx=1);for(e=2,if(n>1,vecmax(f[,2])), mx=max(mx,prod(i=1,#f[,1],f[i,1]^(f[i,2]\e*e))));mx \\ Charles R Greathouse IV, Dec 28 2011

a(n) = max{ A001597(k) : A001597(k)|n }. - R. J. Mathar, Jun 09 2016

Values matching definition restored by Franklin T. Adams-Watters, Jun 06 2016

A338539 Numbers having exactly two non-unitary prime factors.

Original entry on oeis.org

36, 72, 100, 108, 144, 180, 196, 200, 216, 225, 252, 288, 300, 324, 360, 392, 396, 400, 432, 441, 450, 468, 484, 500, 504, 540, 576, 588, 600, 612, 648, 675, 676, 684, 700, 720, 756, 784, 792, 800, 828, 864, 882, 936, 968, 972, 980, 1000, 1008, 1044, 1080, 1089

Offset: 1

Amiram Eldar, Nov 01 2020

nonn

Numbers k such that A056170(k) = A001221(A057521(k)) = 2.
Numbers divisible by the squares of exactly two distinct primes.
Subsequence of A036785 and first differs from it at n = 44.
The asymptotic density of this sequence is (3/Pi^2)*(eta_1^2 - eta_2) = 0.0532928864..., where eta_j = Sum_{p prime} 1/(p^2-1)^j (Pomerance and Schinzel, 2011).

			36 = 2^2 * 3^2 is a term since it has exactly 2 prime factors, 2 and 3, that are non-unitary.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Carl Pomerance and Andrzej Schinzel, Multiplicative Properties of Sets of Residues, Moscow Journal of Combinatorics and Number Theory, Vol. 1, No. 1 (2011), pp. 52-66. See pp. 61-62.

Subsequence of A013929 and A036785.
Cf. A001221, A056170, A057521, A190641, A338540, A338541, A338542, A347892 (subsequence).
Cf. A154945 (eta_1), A324833 (eta_2).

Select[Range[1000], Count[FactorInteger[#][[;;,2]], _?(#1 > 1 &)] == 2 &]

A360720 a(n) is the sum of unitary divisors of n that are powerful (A001694).

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 9, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 9, 26, 1, 28, 5, 1, 1, 1, 33, 1, 1, 1, 50, 1, 1, 1, 9, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 28, 1, 9, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 90, 1, 1, 26, 5, 1, 1, 1, 17, 82

Offset: 1

Amiram Eldar, Feb 18 2023

The number of these divisors is given by A323308.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^(3/2) for n = 1..10^8.
Index entries for sequences related to divisors of numbers.

Cf. A001694, A004709, A034444, A034448, A057521, A077610, A092261, A323308.

Similar sequences: A183097, A360722.

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

for(n=1, 100, print1(direuler(p=2, n, (1 - p^3*X^4 - p^2*X^3 + p^3*X^3) / ((1 - X) * (1 - p^2*X^2)))[n], ", ")) \\ Vaclav Kotesovec, Feb 18 2023

Multiplicative with a(p) = 1 and a(p^e) = p^e + 1 for e > 1.
a(n) <= A034448(n), with equality if and only if n is powerful (A001694).
a(n) <= A183097(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s)*zeta(s-1)*Product_{p prime} (1 - p^(1-s) + p^(2-2*s) - p^(2-3*s)).
From Vaclav Kotesovec, Feb 18 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{primes p} (1 - p^(3-4*s) - p^(2-3*s) + p^(3-3*s)).
Sum_{k=1..n} a(k) ~ c * zeta(3/2) * n^(3/2) / 3, where c = Product_{primes p} (1 + 1/p^(3/2) - 1/p^(5/2) - 1/p^3) = 1.48039182258752809541724060173644... (End)
a(n) = A034448(A057521(n)) (the sum of unitary divisors of the powerful part of n). - Amiram Eldar, Dec 12 2023
a(n) = A034448(n)/A092261(n). - Amiram Eldar, Jun 19 2025

A212180 Number of distinct second signatures (cf. A212172) represented among divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3

Offset: 1

Matthew Vandermast, Jun 04 2012

nonn

Completely determined by the exponents >=2 in the prime factorization of n (cf. A212172, A212173).

The fraction of the divisors of n which have a given second signature {S} is also a function of n's second signature. For example, if n has second signature {3,2}, it follows that 1/3 of n's divisors are squarefree. Squarefree numbers are represented with 0's in A212172, in accord with the usual OEIS custom of using 0 for nonexistent elements; in comments, their second signature is represented as { }.

			The divisors of 72 represent a total of 5 distinct second signatures (cf. A212172), as can be seen from the exponents >= 2, if any, in the canonical prime factorization of each divisor:
{ }: 1, 2 (prime), 3 (prime), 6 (2*3)
{2}: 4 (2^2), 9 (3^2), 12 (2^2*3), 18 (2*3^2)
{3}: 8 (2^3), 24 (2^3*3)
{2,2}: 36 (2^2*3^2)
{3,2}: 72 (2^3*3^2)
Hence, a(72) = 5.

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

Cf. A212172, A085082, A088873, A181796, A182860, A212173, A212642, A212643, A212644.

Array[Length@ Union@ Map[Sort@ Select[FactorInteger[#][[All, -1]], # >= 2 &] &, Divisors@ #] &, 88] (* Michael De Vlieger, Jul 19 2017 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); } \\ This function from Charles R Greathouse IV, Aug 13 2013
A212173(n) = A046523(A057521(n));
A212180(n) = { my(vals = Set()); fordiv(n, d, vals = Set(concat(vals, A212173(d)))); length(vals); }; \\ Antti Karttunen, Jul 19 2017

from sympy import factorint, divisors, prod
def P(n): return sorted(factorint(n).values())
def a046523(n):
    x=1
    while True:
        if P(n)==P(x): return x
        else: x+=1
def a057521(n): return 1 if n==1 else prod(p**e for p, e in factorint(n).items() if e != 1)
def a212173(n): return a046523(a057521(n))
def a(n):
    l=[]
    for d in divisors(n):
        x=a212173(d)
        if not x in l:l+=[x, ]
    return len(l)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017

A357669 a(n) is the number of divisors of the powerful part of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 08 2022

The corresponding sum of divisors of the powerful part of n is A295294.

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

Cf. A000005, A001694, A003557, A005117, A056671, A057521, A064549, A295294.

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

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

a(n) = A000005(A057521(n)).
a(n) = A000005(n)/A056671(n).
a(n) = A000005(A064549(A003557(n))).
a(n) = 1 iff n is squarefree (A005117).
a(n) = A000005(n) iff n is powerful (A001694).
Multiplicative with a(p^e) = 1 if e = 1 and e+1 otherwise.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^3 - p^2 + 2*p - 1)/(p^2*(p - 1))) = 2.71098009471568319328... .
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 2/p^(2*s) - 1/p^(3*s)). - Amiram Eldar, Sep 09 2023

cp's OEIS Frontend

A384048 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is squarefree.

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A384050 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a powerful number.

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A360540 a(n) is the cubefull part of n: the largest divisor of n that is a cubefull number (A036966).

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A195086 Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2.

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A071773 a(n) = gcd(rad(n), n/rad(n)), where rad(n) = A007947(n) is the squarefree kernel of n.

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A203025 Largest perfect power divisor of n.

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A338539 Numbers having exactly two non-unitary prime factors.

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