A286708 -id:A286708 - cp's OEIS frontend

A332785 Nonsquarefree numbers that are not squareful.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 204, 207, 208, 212, 220, 224

Offset: 1

Bernard Schott, Feb 24 2020

Sometimes nonsquarefree numbers are misnamed squareful numbers (see 1st comment of A013929). Indeed, every squareful number > 1 is nonsquarefree, but the converse is false. This sequence = A013929 \ A001694 and consists of these counterexamples.
This sequence is not a duplicate: the first 16 terms (<= 68) are the same first 16 terms of A059404, A323055, A242416 and A303946, then 72 is the 17th term of these 4 sequences. Also, the first 37 terms (<= 140) are the same first 37 terms of A317616 then 144 is the 38th term of this last sequence.
From Amiram Eldar, Sep 17 2023: (Start)
Called "hybrid numbers" by Jakimczuk (2019).
These numbers have a unique representation as a product of two numbers > 1, one is squarefree (A005117) and the other is powerful (A001694).
Equivalently, numbers k such that A055231(k) > 1 and A057521(k) > 1.
Equivalently, numbers that have in their prime factorization at least one exponent that is equal to 1 and at least one exponent that is larger than 1.
The asymptotic density of this sequence is 1 - 1/zeta(2) (A229099). (End)

			18 = 2 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is not squareful because 2 divides 18 but 2^2 does not divide 18, hence 18 is a term.
72 = 2^3 * 3^2 is nonsquarefree as it is divisible by the square 3^2, but it is also squareful because primes 2 and 3 divide 72, and 2^2 and 3^2 divide also 72, so 72 is not a term.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Powerful Numbers Multiple of a Set of Primes and Hybrid Numbers, 2019.

Cf. A005117 (squarefree), A013929 (nonsquarefree), A001694 (squareful), A052485 (not squareful).

Cf. A059404, A126706, A229099, A242416, A286708, A303946, A317616, A323055 (first terms are the same).

filter:= proc(n) local F;
 F:= ifactors(n)[2][..,2];
 max(F) > 1 and min(F) = 1
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 15 2024

Select[Range[225], Max[(e = FactorInteger[#][[;;,2]])] > 1 && Min[e] == 1 &] (* Amiram Eldar, Feb 24 2020 *)

isok(m) = !issquarefree(m) && !ispowerful(m); \\ Michel Marcus, Feb 24 2020

from math import isqrt
from sympy import mobius, integer_nthroot
def A332785(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, j = n-1+squarefreepi(integer_nthroot(x,3)[0])+squarefreepi(x), 0, 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)
        return c-l
    return bisection(f,n,n) # Chai Wah Wu, Sep 14 2024

This sequence is A126706 \ A286708.

Sum_{n>=1} 1/a(n)^s = 1 + zeta(s) - zeta(s)/zeta(2*s) - zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1. - Amiram Eldar, Sep 17 2023

A303606 Powers of composite squarefree numbers that are not squarefree.

Original entry on oeis.org

36, 100, 196, 216, 225, 441, 484, 676, 900, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 1764, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4356, 4761, 4900, 5476, 5929, 6084, 6724, 7225, 7396, 7569, 7776, 8281, 8649, 8836, 9025, 9261, 10000, 10404, 10648, 11025, 11236

Offset: 1

Ilya Gutkovskiy, Apr 26 2018

nonn

			196 is in the sequence because 196 = 2^2*7^2.
4900 is in the sequence because 4900 = 2^2*5^2*7^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power.
Eric Weisstein's World of Mathematics, Squarefree.

Intersection of A024619 and A072777.
Intersection of A072774 and A126706.
Intersection of A013929 and A182853.
Cf. A000469, A001597, A005117, A120944, A286708.

Select[Range[12000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && ! SquareFreeQ[#] && ! PrimePowerQ[#] &]
seq[max_] := Module[{sp = Select[Range[Floor@Sqrt[max]], SquareFreeQ[#] && PrimeNu[#] > 1 &], s = {}}, Do[s = Join[s, sp[[k]]^Range[2, Floor@Log[sp[[k]], max]]], {k, 1, Length[sp]}]; Union@s]; seq[10^4] (* Amiram Eldar, Feb 12 2021 *)

from math import isqrt
from sympy import mobius, primepi, integer_nthroot
def A303606(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    def f(x): return n-3+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(2,y))
    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 19 2024

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A120944(n)-1)*A120944(n)) = Sum_{k>=2} (zeta(k)/zeta(2*k) - P(k) - 1) = 0.07547719891508850482..., where P(k) is the prime zeta function. - Amiram Eldar, Feb 12 2021

A364054 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod prime(n-1)).

Original entry on oeis.org

1, 3, 6, 11, 4, 15, 2, 19, 38, 61, 32, 63, 26, 67, 24, 71, 18, 77, 16, 83, 12, 85, 164, 81, 170, 73, 174, 277, 384, 57, 283, 29, 160, 23, 162, 13, 315, 158, 321, 154, 327, 148, 329, 138, 331, 134, 333, 122, 345, 118, 347, 114, 353, 112, 363, 106, 369, 100, 371, 94, 375, 92, 385

Offset: 1

Ali Sada, Oct 19 2023

5 is the smallest positive integer missing from the first 1000 terms. Also in the interval a(100) to a(1000) there are no entries less than 100. (From W. Edwin Clark via SeqFan.)
Comments from N. J. A. Sloane, Oct 22 2023 (Start)
It appears that the graph of this sequence is dominated by pairs of diverging lines, as suggested by the sketch (see link). For example, around step n = 4619, a descending line is changing to a descending line around a(4619) = 65, a companion ascending line is coming to an end near a(4594) = 44518, and a strong ascending line is starting up around a(4620) = 88899.
It would be nice to have more terms, in order to get better estimates of the times t_i where these transitions happen, and heights alpha_i, beta_i, gamma_i where line breaks are.
The only well-defined points are the (t_i, alpha_i) where the descending lines end, as can be seen from the b-file, where the end point a(4619) = 65 is well-defined. The other transitions, where an ascending line changes to a descending line, are less obvious.  It would be nice to know more.
Can the t_i and alpha_i sequences be traced back to the start of the sequence? Of course the alpha_i sequence is not monotonic, and in particular we do not know at present if some alpha_i is equal to 5.
(End)
a(28149) = 7. - Chai Wah Wu, Oct 22 2023
Comment from N. J. A. Sloane, Mar 05 2024 (Start):
At present there is no OEIS entry for the inverse sequence, since it is not known if 5 appears here.
The initial values of the inverse sequence are
n.....1..2..3..4..5..6....7.....8..9..10..11... . . .
index.1..7..2..5..?..3..28149..81..?...?...4... . . . (End)

			For n = 2, prime(2-1) = prime(1) = 2; a(1) = 1, so a(1) mod 2 = 1, so a(2) is the least positive integer == 1 (mod 2) that has not yet appeared; 1 has appeared, so a(2) = 3.
For n = 3, prime(3-1) = 3; a(2) mod 3 = 0, so a(3) is the least unused integer == 0 mod 3, which is 6, so a(3) =  6.
For n = 4, prime(4-1) = 5; a(3) mod 5 = 1, and 6 has already been used, so a(4) = 11.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Sketch showing the main features of the graph
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, 2048 X 2048 raster showing a(n) , n = 1..4194304 in rows of 2048 terms, left to right, then continued below for 2048 rows total. Color indicates terms as follows: black = empty product {1}, red = prime (A40), gold = composite prime power (A246547), bright green = primorial A2110(k), k > 1, light green = squarefree semiprime (A6881 \ {6}), dark green = squarefree composite (A120944 \ {A2110 U A6881}), blue = numbers neither squarefree nor composite (A126706 \ A286708 = A332785), purple = squareful numbers that are not prime powers (A286708).
Chai Wah Wu, Graph of first 10^8 terms

Cf. A006509, A125717.
For a(n-1) (mod prime(n-1)) see A366470.
Records: A368384, A368385.
See also A366475, A366477.

a[1] = 1; a[n_] := a[n] = Module[{p = Prime[n - 1], k = 2, s = Array[a, n - 1]}, While[! FreeQ[s, k] || ! Divisible[k - a[n - 1], p], k++]; k]; Array[a, 100] (* Amiram Eldar, Oct 20 2023 *)
nn = 2^20; c[] := False; m[] := 0; a[1] = j = 1; c[0] = c[1] = True;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++];
    Set[{a[n], c[k], j}, {k, True, k}], {n, 2, nn}], n];
Array[a, nn] (* Michael De Vlieger, Oct 26 2023, fast, based on congruence, avoids search *)

from itertools import count, islice
from sympy import nextprime
def A364054_gen(): # generator of terms
    a, aset, p = 1, {0,1}, 2
    while True:
        yield a
        for b in count(a%p,p):
            if b not in aset:
                aset.add(b)
                a, p = b, nextprime(p)
                break
A364054_list = list(islice(A364054_gen(),30)) # Chai Wah Wu, Oct 22 2023

A361098 Intersection of A360765 and A360768.

Original entry on oeis.org

36, 48, 50, 54, 72, 75, 80, 96, 98, 100, 108, 112, 135, 144, 147, 160, 162, 189, 192, 196, 200, 216, 224, 225, 240, 242, 245, 250, 252, 270, 288, 294, 300, 320, 324, 336, 338, 350, 352, 360, 363, 375, 378, 384, 392, 396, 400, 405, 416, 432, 441, 448, 450, 468, 480, 484, 486, 490, 500, 504, 507, 525

Offset: 1

Michael De Vlieger, Mar 15 2023

nonn

Numbers k that are neither prime powers nor squarefree, such that rad(k) * A053669(k) < k and k/rad(k) >= A119288(k), where rad(k) = A007947(k).
Numbers k such that A360480(k), A360543(k), A361235(k), and A355432(k) are positive.
Subset of A126706. All terms are neither prime powers nor squarefree.
From Michael De Vlieger, Aug 03 2023: (Start)
Superset of A286708 = A001694 \ {{1} U A246547}, which in turn is a superset of A303606. We may write k in A286708 as m*rad(k)^2, m >= 1. Since omega(k) > 1, it is clear both k/rad(k) > A053669(k) and k/rad(k) >= A119288(k). Also superset of A359280 = A286708 \ A303606.
This sequence contains {A002182 \ A168263}. (End)

			For prime p, A360480(p) = A360543(p) = A361235(p) = A355432(p) = 0, since k < p is coprime to p.
For prime power n = p^e > 4, e > 0, A360543(n) = p^(e-1) - e, but A360480(n) = A361235(n) = A355432(n) = 0, since the other sequences require omega(n) > 1.
For squarefree composite n, A360480(n) >= 1 and A361235(n) >= 1 (the latter for n > 6), but A360543(n) = A355432(n) = 0, since the other sequences require at least 1 prime power factor p^e | n with e > 0.
For n = 18, A360480(n) = | {10, 14, 15} | = 3,
            A360543(n) = | {} | = 0,
            A361235(n) = | {4, 8, 16} | = 3,
            A355432(n) = | {12} | = 1.
Therefore 18 is not in the sequence.
For n = 36, A360480(n) = | {10, 14, 15, 20, 21, 22, 26, 28, 33, 34} | = 10,
            A360543(n) = | {30} | = 1,
            A361235(n) = | {8, 16, 27, 32} | = 4,
            A355432(n) = | {24} | = 1.
Therefore 36 is the smallest term in the sequence.
Table pertaining to the first 12 terms:
Key: a = A360480, b = A360543, c = A243823; d = A361235, e = A355432, f = A243822;
g = A046753 = f + c, tau = A000005, phi = A000010.
    n |  a + b =  c | d + e = f | g + tau + phi - 1 =  n
  ------------------------------------------------------
   36 | 10 + 1 = 11 | 4 + 1 = 5 | 16 +  9 + 12 - 1 =  36
   48 | 16 + 2 = 18 | 3 + 2 = 5 | 23 + 10 + 16 - 1 =  48
   50 | 18 + 1 = 19 | 4 + 2 = 6 | 25 +  6 + 20 - 1 =  50
   54 | 19 + 2 = 21 | 4 + 4 = 8 | 29 +  8 + 18 - 1 =  54
   72 | 27 + 4 = 31 | 4 + 2 = 6 | 37 + 12 + 24 - 1 =  72
   75 | 25 + 2 = 27 | 2 + 1 = 3 | 30 +  6 + 40 - 1 =  75
   80 | 32 + 3 = 35 | 3 + 1 = 4 | 39 + 10 + 32 - 1 =  80
   96 | 38 + 7 = 45 | 4 + 4 = 8 | 53 + 12 + 32 - 1 =  96
   98 | 41 + 3 = 44 | 5 + 2 = 7 | 51 +  6 + 42 - 1 =  98
  100 | 42 + 4 = 46 | 4 + 2 = 6 | 52 +  9 + 40 - 1 = 100
  108 | 44 + 8 = 52 | 5 + 4 = 9 | 61 + 12 + 36 - 1 = 108
  112 | 48 + 3 = 51 | 3 + 1 = 4 | 55 + 10 + 48 - 1 = 112

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Diagram showing k = 1..n for n = 1..54 in blue for k counted by A360480(n), in green for k counted by A360543(n), in gold for k counted by A361235(n), and in magenta for k counted by A355432(n). Red dots indicate k | n such that k > 1, while gray dots indicate gcd(k, n) = 1.
Michael De Vlieger, 1016 X 1016 pixel bitmap read left to right in rows, then top to bottom where the k-th pixel is black if A126706(k) is in this sequence, else white (1032256 pixels total).

Cf. A000005, A000010, A001694, A002182, A007947, A045763, A053669, A119288, A126706, A168263, A243822, A243823, A355432, A360480, A360543, A361235.

Subsets: A095990\{18}, A286708, A303606, A359280.

nn = 2^16;
a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p];
s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
Reap[ Do[n = s[[j]];
    If[And[#1*a053669[n] < n, n/#1 >= #2] & @@ {Times @@ #, #[[2]]} &@
      FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}]][[-1, -1]]

A376936 Powerful numbers divisible by cubes of 2 distinct primes.

Original entry on oeis.org

216, 432, 648, 864, 1000, 1296, 1728, 1944, 2000, 2592, 2744, 3375, 3456, 3888, 4000, 5000, 5184, 5400, 5488, 5832, 6912, 7776, 8000, 9000, 9261, 10000, 10125, 10368, 10584, 10648, 10800, 10976, 11664, 13500, 13824, 15552, 16000, 16200, 16875, 17496, 17576, 18000

Offset: 1

Michael De Vlieger, Oct 16 2024

Numbers m with coreful divisors d, m/d such that neither d | m/d nor m/d | d, i.e., numbers m such that there exists a divisor pair (d, m/d) such that rad(d) = rad(m/d) but gcd(d, m/d) > 1 is neither d nor m/d, where rad = A007947. Divisors in each pair must be dissimilar and each in A126706.
Proper subset of A320966.
Contains A372695, A177493, and A162142. Does not contain A085986.

			216 is in the sequence since rad(12) | rad(18), but 12 does not divide 18 and 18 does not divide 12.
432 is a term since rad(18) | rad(24), but 18 does not divide 24 and 24 does not divide 18.
Table of coreful divisors d, a(n)/d such that neither d | a(n)/d nor a(n)/d | d for select a(n)
   n |   a(n)   divisor pairs d X a(n)/d
  ---------------------------------------------------------------------------
   1 |   216:   12 X 18;
   2 |   432:   18 X 24;
   3 |   648:   12 X 54;
   4 |   864:   24 X 36, 18 X 48;
   5 |  1000:   20 X 50;
   6 |  1296:   24 X 54;
   7 |  1728:   18 X 96, 36 X 48;
   8 |  1944:   12 X 162, 36 X 54;
   9 |  2000:   40 X 50;
  10 |  2592:   24 X 108, 48 X 54;
  11 |  2744:   28 X 98;
  12 |  3375:   45 X 75;
  13 |  3456:   18 X 192, 36 X 96, 48 X 72;
  22 |  7776:   24 X 324, 48 X 162, 54 X 144, 72 X 108;
  58 | 31104:   48 X 648, 54 X 576, 96 X 324, 108 X 288, 144 X 216, 162 X 192

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Notes on this sequence

Cf. A007947, A030078, A085986, A126706, A162142, A177493, A286708, A307958, A320966, A361430, A370329, A372695, A376514.

Cf. A082020, A082695.

Union@ Select[
  Flatten@ Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[#/b^3]}] &[20000],
  Length@ Select[FactorInteger[#][[All, -1]], # > 2 &] >= 2 &]

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - (15/Pi^2) * (1 + Sum_{prime} 1/((p-1)*(p^2+1))) = 0.021194288968234037106579437374641326044... . - Amiram Eldar, Nov 08 2024

A369374 Powerful numbers k that have a primorial kernel and more than 1 distinct prime factor.

Original entry on oeis.org

36, 72, 108, 144, 216, 288, 324, 432, 576, 648, 864, 900, 972, 1152, 1296, 1728, 1800, 1944, 2304, 2592, 2700, 2916, 3456, 3600, 3888, 4500, 4608, 5184, 5400, 5832, 6912, 7200, 7776, 8100, 8748, 9000, 9216, 10368, 10800, 11664, 13500, 13824, 14400, 15552, 16200

Offset: 1

Michael De Vlieger, Jan 22 2024

nonn

Numbers k such that Omega(k) > omega(k) > 1 with all prime power factors p^m for m > 1, such that squarefree kernel rad(k) is in A002110, where Omega = A001222, omega = A001221, and rad(k) = A007947(k).
Union of the product of the squares of primorials P(n)^2, n > 1, and the set of prime(n)-smooth numbers.
Superset of A364930.
Proper subset of A367268, which in turn is a proper subset of A126706.

			This sequence is the union of the following infinite sets:
P(2)^2 * A003586 = {36, 72, 108, 144, 216, 288, 324, ...}
                 = { m*P(2)^2 : rad(m) | P(2) }.
P(3)^2 * A051037 = {900, 1800, 2700, 3600, 4500, 5400, ...}
                 = { m*P(3)^2 : rad(m) | P(3) }.
P(4)^2 * A002473 = {44100, 88200, 132300, 176400, ...}
                 = { m*P(4)^2 : rad(m) | P(4) }, etc.

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

Cf. A001221, A001222, A001694, A002110, A007947, A055932, A126706, A286708, A364930, A367268, A369417.

With[{nn = 2^14},
  Select[
    Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
      Not@*PrimePowerQ],
    And[EvenQ[#],
      Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}] &] ]

{a(n)} = { m*P(n)^2 : P(n) = Product_{j = 1..n} prime(n), rad(m) | P(n), n > 1 }.
Intersection of A286708 and A055932.
A286708 is the union of A369417 and this sequence.

A364930 Products of primorials that are squareful but not prime powers.

Original entry on oeis.org

36, 72, 144, 216, 288, 432, 576, 864, 900, 1152, 1296, 1728, 1800, 2304, 2592, 3456, 3600, 4608, 5184, 5400, 6912, 7200, 7776, 9216, 10368, 10800, 13824, 14400, 15552, 18432, 20736, 21600, 27000, 27648, 28800, 31104, 32400, 36864, 41472, 43200, 44100, 46656, 54000

Offset: 1

Michael De Vlieger, Dec 12 2023

Proper subset of A055932.
Proper subset of A364710; contains k in {A025487 \ {A000079 U A002110}} that are not in A332785.
The only highly composite term is 36.

Michael De Vlieger, Table of n, a(n) for n = 1..15402 (All terms k <= A002110(16))

Cf. A000079, A001694, A002110, A025487, A126706, A286708, A332785, A364710.

(* Load May 19 2018 function f at A025487, then run the following: *)
Select[Union@ Flatten@ f[k], And[PrimeOmega[#] > PrimeNu[#] > 1, AllTrue[FactorInteger[#][[All, -1]], # > 1 &] ] &]

This sequence is A364710 \ A332785.

Intersection of A025487 and A286708.

A364702 Numbers k in A361098 that are not divisible by A007947(k)^2.

Original entry on oeis.org

48, 50, 54, 75, 80, 96, 98, 112, 135, 147, 160, 162, 189, 192, 224, 240, 242, 245, 250, 252, 270, 294, 300, 320, 336, 338, 350, 352, 360, 363, 375, 378, 384, 396, 405, 416, 448, 450, 468, 480, 486, 490, 504, 507, 525, 528, 540, 550, 560, 567, 578, 588, 594, 600

Offset: 1

Michael De Vlieger, Aug 03 2023

nonn

Subset of A126706, the set of numbers k neither prime powers nor squarefree, i.e., k such that A001222(k) > A001221(k) > 1.
Let p = A119288(k) be the second smallest prime factor of k. Let q = A053669(k) be the smallest prime that does not divide k. Let r = rad(k) = A007947(k) be the squarefree kernel of k. Define sequence S = A361098 = {k : Omega(k) > omega(k) > 1, q*r < k, p*r <= k} = A361098.
Sequence T = A286708 represents numbers in A001694 that are not prime powers. Numbers k in T are such that k = m*r^2, m >= 1, by definition. Since we may rewrite q*r < k instead as q*r < m*r^2, it is clear since omega(r) > 1, that q < r. Further, we may rewrite p*r <= k instead as p*r <= m*r^2, and since p | r, p < r as omega(r) > 1, we see that S contains T.
This sequence gives k that are in S but not in T.

			Let B = A126706.
B(1) = 12 is not in the sequence since 3*6 > 12.
B(2) = 18 is not in the sequence, since, though 3*6 = 18, 5*6 > 18.
B(6) = S(1) = 36 is not in the sequence since, though 3*6 < 36 and 5*6 < 36, rad(36)^2 = 6^2 | 36, hence B(6) = T(1).
B(10) = S(2) = a(1) = 48 is in the sequence since rad(48) = 6, and 6^2 does not divide 48.
B(11) = S(3) = a(2) = 50 is in the sequence since rad(50) = 10, and 10^2 does not divide 50, etc.

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

Cf. A001221, A001222, A001694, A007947, A053669, A119288, A126706, A286708, A361098.

nn = 2^10; a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p]; s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]; Reap[Do[n = s[[j]]; If[And[#1*a053669[n] < n, #1*#2 <= n, ! Divisible[n, #1^2]] & @@ {Times @@ #, #[[2]]} &@ FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}] ][[-1, -1]]

This sequence is A361098 \ A286708.

A365308 Powers of primorials P(k)^m, k > 1, m > 1, where P(k) = A002110(k).

Original entry on oeis.org

36, 216, 900, 1296, 7776, 27000, 44100, 46656, 279936, 810000, 1679616, 5336100, 9261000, 10077696, 24300000, 60466176, 362797056, 729000000, 901800900, 1944810000, 2176782336, 12326391000, 13060694016, 21870000000, 78364164096, 260620460100, 408410100000, 470184984576

Offset: 1

Michael De Vlieger, Oct 02 2023

Proper subset of A303606, in turn a proper subset of A286708, in turn a proper subset of A126706.

Numbers in A322793 that are not powers of 2.

			Terms less than 10^4 include P(2)^2 = 36, P(2)^3 = 216, P(2)^4 = 1296, P(2)^5 = 7776, and P(3)^2 = 900.

Michael De Vlieger, Table of n, a(n) for n = 1..4935
Michael De Vlieger, 1024 X 1024 Bitmap showing A322793(n) in black if a power of 2 (i.e., in A000079) else white if in this sequence, n = 1..2^20, arranged from left to right in rows, then from top to bottom.

Cf. A000079, A002110, A100778, A126706, A286708, A303606, A322793, A325374.

nn = 2^39; k = 2; P = 6; Union@ Reap[While[j = 2; While[P^j < nn, Sow[P^j]; j++]; j > 2, k++; P *= Prime[k]] ][[-1, 1]]

Intersection of A100778 and A303606.
This sequence is {A325374 \ {A002110 \ {1,2}}} = {A322793 \ {A000079 \ {1,2}}}.
Sum_{n>=1} 1/a(n) = Sum_{k>=2} 1/(P(k)*(P(k)-1)) = 0.03450573145072369022... . - Amiram Eldar, Mar 10 2024

A368682 Products of primorials that are perfect powers but not prime powers.

Original entry on oeis.org

36, 144, 216, 576, 900, 1296, 1728, 2304, 3600, 5184, 7776, 9216, 13824, 14400, 20736, 27000, 32400, 36864, 44100, 46656, 57600, 82944, 110592, 129600, 147456, 176400, 186624, 216000, 230400, 248832, 279936, 331776, 373248, 518400, 589824, 705600, 746496, 810000

Offset: 1

Michael De Vlieger, Jan 02 2024

nonn

Intersection of A025487 and A131605.
Proper subset of A286708.
Contains A365308 (perfect powers of composite primorials) and A368508 (perfect powers of composite superprimorials).
These numbers are perfect powers of some smaller product of primorials.

			b(n) = A025487(n).
a(1) = b(11) = 36 = 6^2 = b(4)^2,
a(2) = b(19) = 144 = 12^2 = b(6)^2,
a(3) = b(23) = 216 = 6^3 = b(4)^3,
a(4) = b(33) = 576 = 24^2 = b(8)^2,
a(5) = b(38) = 900 = 30^2 = b(9)^2, etc.

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

Cf. A000079, A000961, A001597, A002110, A025487, A131605, A286708, A365308, A368508, A368681.

Select[Range[36, 2^18, 2], And[Union@ Differences@ PrimePi@ #1 == {1}, AllTrue[Union@ Differences@ #2, # <= 0 &], GCD @@ #2 > 1] & @@ Transpose@ FactorInteger[#] &]

This sequence is { A368681 \ A000079 }.

cp's OEIS Frontend

A332785 Nonsquarefree numbers that are not squareful.

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A303606 Powers of composite squarefree numbers that are not squarefree.

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A364054 a(1) = 1; for n > 1, a(n) is the least positive integer not already in the sequence such that a(n) == a(n-1) (mod prime(n-1)).

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A361098 Intersection of A360765 and A360768.

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A376936 Powerful numbers divisible by cubes of 2 distinct primes.

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A369374 Powerful numbers k that have a primorial kernel and more than 1 distinct prime factor.

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A364930 Products of primorials that are squareful but not prime powers.

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A364702 Numbers k in A361098 that are not divisible by A007947(k)^2.