A052485 -id:A052485 - cp's OEIS frontend

A351982 Number of integer partitions of n into prime parts with prime multiplicities.

1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 3, 0, 1, 1, 3, 3, 3, 0, 1, 4, 5, 5, 3, 3, 5, 8, 5, 5, 6, 8, 8, 11, 7, 8, 10, 17, 14, 14, 12, 17, 17, 21, 18, 23, 20, 28, 27, 31, 27, 36, 32, 35, 37, 46, 41, 52, 45, 60, 58, 63, 59, 78, 71, 76, 81, 87, 80, 103, 107, 113, 114, 127

Offset: 0

Gus Wiseman, Mar 18 2022

nonn

			The partitions for n = 4, 6, 10, 19, 20, 25:
  (22)  (33)   (55)     (55333)     (7733)       (55555)
        (222)  (3322)   (55522)     (77222)      (77722)
               (22222)  (3333322)   (553322)     (5533333)
                        (33322222)  (5522222)    (5553322)
                                    (332222222)  (55333222)
                                                 (55522222)
                                                 (333333322)
                                                 (3333322222)

The version for just prime parts is A000607, ranked by A076610.
The version for just prime multiplicities is A055923, ranked by A056166.
For odd instead of prime we have A117958, ranked by A352142.
The constant case is A230595, ranked by A352519.
Allowing any multiplicity > 1 gives A339218, ranked by A352492.
These partitions are ranked by A346068.
The non-constant case is A352493, ranked by A352518.
A000040 lists the primes.
A001221 counts constant partitions of prime length, ranked by A053810.
A001694 lists powerful numbers, counted A007690, weak A052485.
A038499 counts partitions of prime length.
A101436 counts parts of prime signature that are themselves prime.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, sum A001222.
A257994 counts prime indices that are prime, nonprime A330944.
Cf. A000720, A000961, A001156, A011757, A052335, A164336, A320628.

Table[Length[Select[IntegerPartitions[n], And@@PrimeQ/@#&&And@@PrimeQ/@Length/@Split[#]&]],{n,0,30}]

A062760 a(n) is n divided by the largest power of the squarefree kernel of n (A007947) which divides it.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 1, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 16, 1, 7, 3, 1, 1, 1, 1, 4

Offset: 1

Labos Elemer, Jul 16 2001

nonn

a(n) divides A003557 but is not equal to it.

a(n) is least d such that the prime power exponents of n/d are all equal; see also A066636. - David James Sycamore, Jun 13 2024

			n=1800: the squarefree kernel is 2*3*5 = 30 and 900 = 30^2 divides n, a(1800) = 2, the quotient of 1800/900.

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

Cf. A001694, A003557, A007947, A051904, A003557, A052485, A062759.
Cf. A059404 (n such that a(n)>1), A072774 (n such that a(n)=1).
Cf. A066636.

f:= proc(n) local F,m,t;
  F:= ifactors(n)[2];
  m:= min(seq(t[2],t=F));
  mul(t[1]^(t[2]-m),t=F)
end proc:
map(f, [$1..200]); # Robert Israel, Nov 03 2017

{1}~Join~Table[n/#^IntegerExponent[n, #] &@ Last@ Select[Divisors@ n, SquareFreeQ], {n, 2, 104}] (* Michael De Vlieger, Nov 02 2017 *)
a[n_] := Module[{f = FactorInteger[n], e}, e = Min[f[[;; , 2]]]; f[[;; , 2]] -= e; Times @@ Power @@@ f]; Array[a, 100] (* Amiram Eldar, Feb 12 2023 *)

A007947(n) = factorback(factorint(n)[, 1]); \\ Andrew Lelechenko, May 09 2014
A051904(n) = if(1==n,0,vecmin(factor(n)[, 2])); \\ After Charles R Greathouse IV's code
A062760(n) = n/(A007947(n)^A051904(n)); \\ Antti Karttunen, Sep 23 2017

a(n) = n/(A007947(n)^A051904(n)).

a(n) = n/A062759(n). - Amiram Eldar, Feb 12 2023

A085971 Union of primes and numbers that are not prime powers (A000040, A024619).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Reinhard Zumkeller, Jul 06 2003

Complement of A025475;
A085972(n) = Max{k: a(k)<=n};
different from A007916 and A052485, as a(28)=36;
A085818(a(n)) = A000040(n).

With[{nn=100},Union[Join[Prime[Range[PrimePi[nn]]],DeleteCases[Range[2,80], ?(PrimePowerQ[#]&)]]]] (* _Harvey P. Dale, May 15 2019 *)

is(n)=isprimepower(n)<2 && n>1 \\ Charles R Greathouse IV, Oct 19 2015

from sympy import primepi, integer_nthroot
def A085971(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 20 2024

a(n) = n + o(sqrt n). - Charles R Greathouse IV, Oct 19 2015

A352492 Powerful numbers whose prime indices are all prime numbers.

Original entry on oeis.org

1, 9, 25, 27, 81, 121, 125, 225, 243, 289, 625, 675, 729, 961, 1089, 1125, 1331, 1681, 2025, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 5625, 6075, 6561, 6889, 7225, 7803, 8649, 9801, 10125, 11881, 11979, 14641, 15125, 15129, 15625, 16129, 16875

Offset: 1

Gus Wiseman, Mar 24 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices (not prime factors) begin:
    1: {}
    9: {2,2}
   25: {3,3}
   27: {2,2,2}
   81: {2,2,2,2}
  121: {5,5}
  125: {3,3,3}
  225: {2,2,3,3}
  243: {2,2,2,2,2}
  289: {7,7}
  625: {3,3,3,3}
  675: {2,2,2,3,3}
  729: {2,2,2,2,2,2}
  961: {11,11}
For example, 675 = prime(2)^3 prime(3)^2 = 3^3 * 5^2.

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

Powerful numbers are A001694, counted by A007690.
The version for prime exponents instead of indices is A056166, counted by A055923.
This is the powerful case of A076610 (products of A006450), counted by A000607.
The partitions with these Heinz numbers are counted by A339218.
A000040 lists primes.
A031368 lists primes of odd index, products A066208.
A101436 counts exponents in prime factorization that are themselves prime.
A112798 lists prime indices, reverse A296150, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A053810 lists all numbers p^q with p and q prime, counted by A230595.
A257994 counts prime indices that are themselves prime, complement A330944.
Cf. A000720, A000961, A001597, A007821, A007916, A052485, A109297, A140319, A164336, A181819, A325131, A330945, A346068.

Select[Range[1000],#==1||And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&&Min@@Last/@FactorInteger[#]>1&]

Intersection of A001694 and A076610.

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/(p*(p-1))) = 1.24410463... - Amiram Eldar, May 04 2022

A062759 Largest power of squarefree kernel of n (= A007947) which divides n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 6, 25, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 49, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 64, 65, 66, 67, 34, 69, 70, 71, 36, 73

Offset: 1

Labos Elemer, Jul 16 2001

nonn

a(n) is a first power if and only if n is not a powerful number (A001694, A052485).

			n = 1800: squarefree kernel is 2*3*5 = 30 and a(1800) = 900 = 30^2 divides n, exponent of 30 is the smallest prime exponent of 1800 = 2*2*2*3*3*5*5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001694, A003557, A007947, A051904, A052485, A062759, A065463.

a062759 n = a007947 n ^ a051904 n  -- Reinhard Zumkeller, Jul 15 2012

{1}~Join~Table[#^IntegerExponent[n, #] &@ Last@ Select[Divisors@ n, SquareFreeQ], {n, 2, 73}] (* Michael De Vlieger, Nov 02 2017 *)
a[n_] := Module[{f = FactorInteger[n], e}, e = Min[f[[;; , 2]]]; f[[;; , 2]] = e; Times @@ Power @@@ f]; Array[a, 100] (* Amiram Eldar, Feb 12 2023 *)

a(n) = {if(n==1, 1, my(f = factor(n), e = vecmin(f[,2])); prod(i = 1, #f~, f[i,1]^e));} \\ Amiram Eldar, Feb 12 2023

a(n) = A007947(n)^A051904(n).
From Amiram Eldar, Feb 12 2023: (Start)
a(n) = n/A062759(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463 / 2 = 0.352221... . (End)

A325240 Numbers whose minimum prime exponent is 2.

Original entry on oeis.org

4, 9, 25, 36, 49, 72, 100, 108, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 675, 676, 784, 800, 841, 900, 961, 968, 972, 1089, 1125, 1152, 1156, 1225, 1323, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1764, 1800

Offset: 1

Gus Wiseman, Apr 15 2019

nonn

Or barely powerful numbers, a subset of powerful numbers A001694.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose minimum multiplicity is 2 (counted by A244515).
Powerful numbers (A001694) that are not cubefull (A036966). - Amiram Eldar, Jan 30 2023

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    9: {2,2}
   25: {3,3}
   36: {1,1,2,2}
   49: {4,4}
   72: {1,1,1,2,2}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}
  169: {6,6}
  196: {1,1,4,4}
  200: {1,1,1,3,3}
  225: {2,2,3,3}
  288: {1,1,1,1,1,2,2}
  289: {7,7}
  324: {1,1,2,2,2,2}
  361: {8,8}
  392: {1,1,1,4,4}
  400: {1,1,1,1,3,3}

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

Positions of 2's in A051904.
Maximum instead of minimum gives A067259.
Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.
Cf. A065483, A082695.

Select[Range[1000],Min@@FactorInteger[#][[All,2]]==2&]

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023

from math import isqrt, gcd
from sympy import integer_nthroot, factorint, mobius
def A325240(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c += integer_nthroot(z//y**4,3)[0]
        return c
    return bisection(f,n,n**2) # Chai Wah Wu, Oct 02 2024

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - Amiram Eldar, Jan 30 2023

A366250 Numbers k that are not powerful and do not have a strictly superior squarefree divisor.

Original entry on oeis.org

48, 54, 96, 160, 162, 192, 224, 250, 320, 375, 384, 405, 448, 486, 567, 640, 686, 704, 768, 832, 896, 960, 1029, 1080, 1200, 1215, 1250, 1280, 1350, 1408, 1440, 1458, 1500, 1536, 1620, 1664, 1701, 1715, 1792, 1875, 1920, 2016, 2058, 2160, 2176, 2250, 2268, 2352

Offset: 1

Peter Munn and Michael De Vlieger, Feb 08 2024

nonn

A number k does not have a strictly superior squarefree divisor if and only if k is at least as large as the square of rad(k), the largest squarefree divisor of k. All powerful numbers (A001694) have this property. This sequence lists the other such numbers.
Let rad(k) = A007947(k), the largest squarefree divisor, i.e., the squarefree kernel of k. A341645 lists the numbers without a strictly superior squarefree divisor.
A341645 = { k : rad(k) <= k/rad(k) } = { k : A007947(k) <= A003557(k) }, and it is evident that rad(k) <= k/rad(k) is true for powerful k, that is, k in A001694.
Since A001694 contains A001597, the above is also true for perfect powers k; A001597 is a proper subset of A341645.
This sequence contains "weak" k (in A052485) such that rad(k) < k/rad(k).
The presence of a number, k, in this sequence depends only upon A290110(k), i.e., upon the factorization pattern of its sequence of divisors as defined in A191743.
Let S = A006939 and let P = A002110. Almost all superprimorials are in this sequence: S \ {1, 2, 12, 360} is a proper subset. S(i) = S(i-1)*P(i), where S(i-1) = A003557(S(i)) and P(i) = rad(S(i)), and for i > 4, S(i-1) > P(i). Since prime(i) | S(i) but prime(i)^2 does not divide S(i), S(i) is not powerful. Corollary: almost all superprimorials are in A341645, since this sequence is a proper subset of A341645.

			Let b(n) = A364702(n).
a(1) = b(1) = 48 since rad(48) < 48/rad(48), 6 < 8.
b(2) = 50 is not in the sequence since rad(50) > 50/rad(50), 10 > 5.
a(2) = b(3) = 54 since 6 < 9, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A002110, A003557, A006939, A007947, A052485, A191743, A290110, A341645, A364702.

Select[Range[2, 2400], And[! AllTrue[#2[[All, -1]], # > 1 &], #1 >= Apply[Times, #2[[All, 1]]^2]] & @@ {#, FactorInteger[#]} &]

isok(m) = if (!ispowerful(m), my(d=divisors(m)); #select(x->(issquarefree(x) && (x^2>m)), d) == 0); \\ Michel Marcus, Feb 11 2024

Set difference of A341645 and A001694.
Intersection of A341645 and A364702 where the latter is a proper subset of A052485.
Sequence contains infinite intersections of A052485 and { k = m*s : s is squarefree, rad(m) | s, 1 < s < m }.
{a(n)} = union of { k = s*m : s > 1 is squarefree, rad(m) | s, m >= s, k is not powerful }.
{a(n)} = { k in A364702 : k >= rad(k)^2 }.

A297404 A binary representation of the positive exponents that appear in the prime factorization of a number, shown in decimal.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 3, 1, 1, 1, 8, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 4, 3, 1, 1, 1, 16, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 3, 1, 1, 3, 32, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 3, 3, 1, 1, 1, 9, 8, 1, 1, 3, 1, 1

Offset: 1

Rémy Sigrist, Dec 29 2017

This sequence is similar to A087207; here we encode the exponents, there the prime numbers appearing in the prime factorization of a number.
The binary representation of a(n) shows which exponents appear in the prime factorization of n, but without multiplicities:
- for any prime number p and k > 0, if p^k divides n but p^(k+1) does not divide n, then a(n) AND 2^(k-1) = 2^(k-1) (where AND denotes the bitwise AND operator),
- conversely, if a(n) AND 2^(k-1) = 2^(k-1) for some k > 0, then there is prime number p such that p^k divides n but p^(k+1) does not divide n.

			For n = 90:
- 90 = 5^1 * 3^2 * 2^1,
- the exponents appearing in the prime factorization of 90 are 1 and 2,
- hence a(90) = 2^(1-1) + 2^(2-1) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A000120, A000695, A001694, A004709, A005117, A038109, A052485, A059404, A087207, A328400.

Array[Total@ Map[2^(# - 1) &, Union[FactorInteger[#][[All, -1]] ]] - Boole[# == 1] &, 86] (* Michael De Vlieger, Dec 29 2017 *)

a(n) = my (x=Set(factor(n)[,2]~)); sum(i=1, #x, 2^(x[i]))/2

a(p^k) = 2^(k-1) for any prime number p and k > 0.
a(n^2) = A000695(2 * a(n)) / 2 for any n > 0.
a(n) <= 1 iff n is squarefree (A005117).
a(n) <= 3 iff n is cubefree (A004709).
a(n) is odd iff n belongs to A052485 (weak numbers).
a(n) is even iff n belongs to A001694 (powerful numbers).
a(n) AND 2 = 2 iff n belongs to A038109 (where AND denotes the bitwise AND operator).
A000120(a(n)) <= 1 iff n belongs to A072774 (powers of squarefree numbers).
A000120(a(n)) > 1 iff n belongs to A059404.
If gcd(m, n) = 1, then a(m * n) = a(m) OR a(n) (where OR denotes the bitwise OR operator).
a(n) = a(A328400(n)). - Peter Munn, Oct 02 2023

A304328 a(n) = n/(largest perfect power divisor of n).

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, 2

Offset: 1

Gus Wiseman, May 10 2018

nonn

Not all terms are squarefree numbers; for example, a(500) = 4.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000961, A001055, A001222, A001597, A001694, A005117, A007716, A007916, A052485, A052486, A059404, A066636, A091050, A160400, A203025, A294068, A303707, A304327, A304339.

Table[n/Last[Select[Divisors[n],#===1||GCD@@FactorInteger[#][[All,2]]>1&]],{n,100}]

a(n)={my(m=1); fordiv(n, d, if(ispower(d), m=max(m,d))); n/m} \\ Andrew Howroyd, Aug 26 2018

a(n) * A203025(n) = n.

A380543 Nonsquarefree weak numbers k whose squarefree kernel is a primorial.

Original entry on oeis.org

12, 18, 24, 48, 54, 60, 90, 96, 120, 150, 162, 180, 192, 240, 270, 300, 360, 384, 420, 450, 480, 486, 540, 600, 630, 720, 750, 768, 810, 840, 960, 1050, 1080, 1200, 1260, 1350, 1440, 1458, 1470, 1500, 1536, 1620, 1680, 1890, 1920, 2100, 2160, 2250, 2400, 2430

Offset: 1

Michael De Vlieger, Jul 15 2025

Numbers in this sequence have the following properties:
The number a(n) is such that rad(a(n))^2 does not divide a(n), i.e., a(n) is not powerful (i.e., in A001694), where rad = A007947.
For i > 1, prime(i) | a(n) implies prime(i-1) | a(n).

			Table of n, a(n) and prime decomposition for n = 1..12:
 n   a(n)  prime decomposition
------------------------------
 1    12   2^2 * 3
 2    18   2   * 3^2
 3    24   2^3 * 3
 4    48   2^4 * 3
 5    54   2   * 3^3
 6    60   2^2 * 3   * 5
 7    90   2   * 3^2 * 5
 8    96   2^5 * 3
 9   120   2^3 * 3   * 5
10   150   2   * 3   * 5^2
11   162   2   * 3^4
12   180   2^2 * 3^2 * 5

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001694, A002110, A007947, A013929, A052485, A055932, A126706, A286708, A332785, A369374, A386223, A386224.

(* Load Fast Mathematica algorithm for A055932 linked at A377854, then: *)
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten[f[6][[3 ;; -1, 2 ;; -1]] ], ! Divisible[#, rad[#]^2] &]

Intersection of A055932 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694.

The union of this sequence and A369374 is A126706.

cp's OEIS Frontend

A351982 Number of integer partitions of n into prime parts with prime multiplicities.

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A062760 a(n) is n divided by the largest power of the squarefree kernel of n (A007947) which divides it.

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A085971 Union of primes and numbers that are not prime powers (A000040, A024619).

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A352492 Powerful numbers whose prime indices are all prime numbers.

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A062759 Largest power of squarefree kernel of n (= A007947) which divides n.

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A325240 Numbers whose minimum prime exponent is 2.

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A366250 Numbers k that are not powerful and do not have a strictly superior squarefree divisor.

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