A332785 -id:A332785 - cp's OEIS frontend

A372404 Powerful k that are not prime powers such that k/rad(k) is nonsquarefree, where rad = A007947.

72, 108, 144, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 675, 784, 800, 864, 968, 972, 1000, 1125, 1152, 1296, 1323, 1352, 1372, 1568, 1600, 1728, 1800, 1936, 1944, 2000, 2025, 2304, 2312, 2500, 2592, 2700, 2704, 2744, 2888, 2916, 3087, 3136, 3200, 3267, 3375, 3456

Offset: 1

Michael De Vlieger, Jun 04 2024

A001694 \ A246547 = A286708, i.e., A286708 contains powerful numbers without perfect prime powers. Hence, this sequence is a proper subset of A286708 which in turn is contained in A126706.

Numbers k in A286708 are such that rad(k)^2 | k. Numbers in this sequence are such that k != A120944(m)^2 for some m, where A120944 is the sequence of squarefree composites.

			The number 36 is not in the sequence since 36/rad(36) = 36/6 = 6, squarefree.
a(1) = 72 since 72/rad(72) = 72/6 = 12 is nonsquarefree.
a(2) = 108 since 108/rad(108) = 108/6 = 18 is nonsquarefree.
a(4) = 200 since 200/rad(200) = 200/10 = 20 is nonsquarefree, etc.

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

Cf. A001694, A007947, A013929, A120944, A126708, A177492, A126708, A246547, A286708, A332785.

With[{nn = 3300},
  Select[
    Select[Rest@ Union@ Flatten@
      Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
    Not@*PrimePowerQ],
  Not@ SquareFreeQ[#/(Times @@ FactorInteger[#][[;;, 1]])] &] ]

rad(n) = factorback(factorint(n)[, 1]);
isok(k) = ispowerful(k) && !isprimepower(k) && !issquarefree(k/rad(k)); \\ Michel Marcus, Jun 05 2024

A286708 = union of A177492 and this sequence.
A001694 = union of A246547, A177492, and this sequence.
A126706 = union of A332785, A177492, and this sequence.

A375934 Numbers whose prime factorization has a second-largest exponent that equals 1.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 03 2024

First differs from A332785 at n = 112: A332785(112) = 360 = 2^3 * 3^2 * 5 is not a term of this sequence.
First differs from A317616 at n = 38: A317616(38) = 144 = 2*4 * 3^2 is not a term of this sequence.
Numbers k such that A375933(k) = 1.
Numbers of the form s1 * s2^e, where s1 and s2 are coprime squarefree numbers that are both larger than 1, and e >= 2.
The asymptotic density of this sequence is Sum_{e>=2} d(e) = 0.36113984820338109927..., where d(e) = Product_{p prime} (1 - 1/p^2 + 1/p^e - 1/p^(e+1)) - Product_{p prime} (1 - 1/p^(e+1)) is the asymptotic density of terms k with A051903(k) = e >= 2.

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

Equals A375142 \ A072777.
Subsequence of A013929.
Subsequences: A054753, A065036, A072357, A095990, A096156, A178739, A178740, A179664, A179668, A179692, A189987, A360793, A375076.
Cf. A001221, A005117, A051903, A051904, A072777, A317616, A332785, A375142, A375933.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[0, Max[Select[e, # < Max[e] &]]] == 1]; Select[Range[300], q]

is(n) = if(n == 1, 0, my(e = factor(n)[,2]); e = select(x -> x < vecmax(e), e); if(#e == 0, 0, vecmax(e) == 1));

A051904(a(n)) = 1.
A051903(a(n)) >= 2.
A001221(a(n)) = 2.

A379545 Triangle read by rows where row n lists powerful divisors d | n (i.e., d in A001694).

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Feb 13 2025

Intersection of row n of A027750 and A001694.

			D(1) = {1} = row 1 of this sequence since 1 | 1 is powerful.
D(2) = {1, 2}; of these, only 1 is powerful.
D(4) = {1, 2, 4}; of these, only 2 is not powerful, so row 4 = {1, 4}.
D(6) = {1, 2, 3, 6}; of these, only 1 is powerful.
D(8) = {1, 2, 4, 8}; of these, only 2 is not powerful, so row 4 = {1, 4, 8}.
D(12) = {1, 2, 3, 4, 6, 12}; of these, only {1, 4} are powerful.
D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, only {1, 4, 9, 36} are powerful, etc.
Table begins:
   n:  row n
  ----------------
   1:  1;
   2:  1;
   3:  1;
   4:  1, 4;
   5:  1;
   6:  1;
   7:  1;
   8:  1, 4, 8;
   9:  1, 9;
  10:  1;
  11:  1;
  12:  1, 4;
  13:  1;
  14:  1;
  15:  1;
  16:  1, 4, 8, 16;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11445 (rows n = 1..6000, flattened).

Cf. A000005, A001694, A005361, A007947, A027750, A332785, A380819.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[Select[Divisors[n], Divisible[#, rad[#]^2] &], {n, 2, 60}] // Flatten

row(n) = select(x -> ispowerful(x), divisors(n)); \\ Amiram Eldar, May 02 2025

First term in row n is 1.
Row n does not contain squarefree factors of n, and also does not contain factors in A332785.
Length of row n = A005361(n) = tau(n/rad(n)), where tau = A000005 and rad = A007947.
For squarefree n, row n = {1}.
Let D(n) = row n of A027750. For prime p and m > 0, row p^m = D(p^m) \ {p}, since d = 1 and p = p^j, j > 1 are powerful, but primes are squarefree (and not powerful).

A379753 Numbers that set records in A379752.

Original entry on oeis.org

60, 120, 240, 480, 840, 1260, 1680, 2520, 3360, 5040, 7560, 10080, 15120, 20160, 27720, 36960, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 443520, 498960, 554400, 665280, 720720, 997920, 1081080, 1330560, 1441440, 2162160, 2882880, 3603600, 4324320, 5765760

Offset: 1

Michael De Vlieger, Jan 01 2025

nonn

Proper subset of the intersection of A025487 and A375055.
Conjecture: subset of A332785 = A126706 \ A286708.
This sequence seems to be rich in highly composite numbers, the prime shape of a(n) resembles that of highly composite numbers, with long tails of large prime factors with multiplicity 1.
Terms not in A002182 are not all of the form 2^5 * prime(i..j), 1 < i < j, for example, a(24) = 443520 = 2^7 * 3^2 * 5 * 7 * 11.

			Let b(n) = A379752(n).
Table showing exponents of prime power factors of a(n) for n = 1..12.
Example: a(6) = 1260 = 2^2 * 3^2 * 5 * 7, hence we write "2.2.1.1".
   n      a(n)       Exp.    b(a(n))
  ----------------------------------
   1       60 **   2.1.1        1   6*10
   2      120 **   3.1.1        2   6*20 = 10*12
   3      240 *    4.1.1        3   6*40 = 10*24 = 12*20
   4      480      5.1.1        4   6*80 = 10*48 = 12*40 = 20*24
   5      840 *    3.1.1.1      6   6*140 = 10*84 = 12*70 = 14*60 = 20*42 = 28*30
   6     1260 *    2.2.1.1      7
   7     1680 *    4.1.1.1      9
   8     2520 **   3.2.1.1     11
   9     3360      5.1.1.1     12
  10     5040 **   4.2.1.1     15
  11     7560 *    3.3.1.1     16
  12    10080 *    5.2.1.1     19
*  = a(n) is highly composite (in A002182),
** = a(n) is superior highly composite (in both A002182 and A002201).

Michael De Vlieger, Table of n, a(n) for n = 1..226
Michael De Vlieger, Prime Power Decomposition of a(n) for n = 1..226, expanding on the table in the example.
Michael De Vlieger, Plot S(n) = P(omega(n))*m at (x,y) = (m, omega(n)), where S is the union of A002182 and this sequence, P is A002110, omega is A001221, and only select m that harbor S(n) shown. Shows the coincidence of many terms in this sequence with A002182. Blue represents m in A002182, gold m in both A002182 and this sequence; dark blue represents m in A002201 (and also in A002182), orange m in both A002201 and this sequence; red indicates terms in this sequence that are not in A002182. Green highlights terms in A002182 but are not determined to be in this sequence.

Cf. A002182, A002201, A025487, A375055, A379752, A379754.

(* Load function f at A025487 *)
r = 0;
s = Select[Union@ Flatten@ f[14][[4 ;; -1]], Not@*SquareFreeQ];
nn = Length[s]; Print[nn];
Reap[Do[k = s[[i]]; If[# > r, r = #; Sow[k]] &@
  Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2] ]] &@ Divisors[k],
    _?(And[1 < GCD @@ {##},
       Nor[Divisible[#2, rad[#1]],
           Divisible[#1, rad[#2]] ] ] & @@ # &)], {i, nn}] ][[-1, 1]]

A386224 Nonsquarefree weak numbers k that are not products of primorials, whose squarefree kernel is a primorial.

Original entry on oeis.org

18, 54, 90, 150, 162, 270, 300, 450, 486, 540, 600, 630, 750, 810, 1050, 1200, 1350, 1458, 1470, 1500, 1620, 1890, 2100, 2250, 2400, 2430, 2940, 3000, 3150, 3240, 3750, 3780, 4050, 4200, 4374, 4410, 4800, 4860, 5250, 5670, 5880, 6000, 6750, 6930, 7290, 7350, 7500

Offset: 1

Michael De Vlieger, Jul 15 2025

			Table of n, a(n) and prime decomposition for n = 1..12:
 n   a(n)  prime decomposition
------------------------------
 1    18   2 * 3^2
 2    54   2 * 3^3
 3    90   2 * 3^2 * 5
 4   150   2 * 3 * 5^2
 5   162   2 * 3^4
 6   270   2 * 3^3 * 5
 7   300   2^2 * 3 * 5^2
 8   450   2 * 3^2 * 5^2
 9   486   2 * 3^5
10   540   2^2 * 3^3 * 5
11   600   2^3 * 3 * 5^2
12   630   2 * 3^2 * 5 * 7

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

Cf. A001694, A002110, A007947, A052485, A025487, A055932, A056808, A126706, A286708, A332785, A369420, A380543, A386223.

(* Load May 19 2018 function f at A025487, then run the following: *)
fQ[x_] :=
 Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &,
   Nest[Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &,
     If[x == 1, {0},
       Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@
         Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ x], 2] ] == x;
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
Select[Union@ Flatten@ f[6][[3 ;; -1, 2 ;; -1]], Nor[Divisible[#, rad[#]^2], fQ[#]] &]

{a(n)} = A380543 \ A386223.
Intersection of A056808 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694, and A056808 = A055932 \ A025487.
The union of this sequence and A369420 is A126706.

A343294 a(n+1) is the smallest preimage k such that A008477(k) = a(n) with a(1) = 100.

Original entry on oeis.org

100, 1024, 625, 33554432, 2116, 70368744177664

Offset: 1

Bernard Schott, Apr 12 2021

Equivalently, when g is the reciprocal map of f = A008477 as defined in the Name, the terms of this sequence are the successive terms of the infinite iterated sequence {m, g(m), g(g(m)), g(g(g(m))), ...} that begins with m = a(1) = 100, hence f(a(n)) = a(n-1).
Why choose 100? Because it is the second integer, after 36, for which there exists a new infinite iterated sequence that begins with g(100) = 1024; then f(100) = 128 with the periodic sequence (128, 49, 128, 49, ...) (see A062307). Explanation: 100 is the 4th nonsquarefree number in A342973 that is also squareful, but the 3 previous such first integers 36, 64, 81 are yet terms of the infinite iterated sequence A343293. Remember that the nonsquarefree terms in A342973 that are not squareful (A332785) have no preimage by f.
When a(n-1) has several preimages by f, as a(n) is the smallest preimage, this sequence is well defined (see examples).
All the terms are nonsquarefree but also powerful, hence they are in A001694.
a(n) < a(n+2) (last comment in A008477) but a(n) < a(n+1) or a(n) > a(n+1).

			a(1) = 100; 1024 = 2^10 so f(1024) = 10^2 = 100: also 5120 = 2^10*5^1 and f(5120) = 10^2*1^5 = 100; we have f(1024) = f(5120) = 100, but as 1024 < 5120, hence g(100) = 1024 and a(2) = 1024.
a(2) = 1024 = f(625) = f(1250), but as 625 < 1250, g(1024) = 625 and a(3) = 625.

Cf. A001694, A008477, A062307, A332785, A342973, A343293.

A367511 Highly composite numbers h(k) = A002182(k) such that h >= rad(h)^2, where rad() = A007947().

Original entry on oeis.org

1, 4, 36, 48, 45360, 50400

Offset: 1

Michael De Vlieger, Feb 08 2024

Alternatively, this sequence lists h(k) such that A301413(k) >= A002110(A108602(k)), where A301413 is the "variable part" v described on page 5 of 12 of the Siano paper.
This sequence is likely finite and full. See Chapter III regarding the structure of "Highly Composite Numbers".
Terms larger than 36 are in A366250; A366250 is in A364702, which is in turn a proper subset of A332785, itself contained in A126706.
36 is in A365308, a proper subset of A303606, contained in A131605, in turn contained in A286708.

			Let P(n) = A002110(n).
a(1) = h(1) = 1 since 1 >= 1^2.
a(2) = h(3) = 4 since 4 >= P(1)^2, 4 >= 2^2.
a(3) = h(7) = 36 since 36 >= P(2)^2, 36 >= 6^2.
a(4) = h(8) = 48 since 48 >= P(2)^2, 48 >= 6^2.
a(5) = h(26) = 43560 since 43560 >= P(4)^2, where P(4) = 210, and 210^2 = 44100.
a(6) = h(27) = 50400 since 50400 >= P(4)^2.
Let V(i) = A301414(i) and let P(j) = A002110(j).
Plot of highly composite h = V(i)*P(j) at (x,y) = (j,i), i = 1..16, j = 1..7, showing h in this sequence in parentheses, and h in A168263 marked with an asterisk (*):
V(i)\P(j) 1   2    6   30   210    2310    30030 ...
        +---------------------------------------
      1 |(1*) 2*   6*
      2 |    (4*) 12*  60*
      4 |         24* 120*  840*
      6 |        (36) 180* 1260*
      8 |        (48) 240  1680*
     12 |             360  2520   27720*
     24 |             720  5040   55440   720720
     36 |                  7560   83160  1081080
     48 |                 10080  110880  1441440
     72 |                 15120  166320  2162160
     96 |                 20160  221760  2882880
    120 |                 25200  277200  3603600
    144 |                        332640  4324320
    216 |                (45360) 498960  6486480
    240 |                (50400) 554400  7207200
    ...

Srinivasa Ramanujan, Highly Composite Numbers, Proc. London Math. Soc. (1916) Vol. 2, No. 14, 347-409.
D. B. Siano and J. D. Siano, An Algorithm for Generating Highly Composite Numbers, 1994.

Cf. A001221, A002110, A002182, A007947, A025487, A108602, A126706, A131605, A168263, A286708, A301413, A301414, A303606, A332785, A365308, A362702, A366250.

(* First load function f at A025487, then run the following: *)
s = Union@ Flatten@ f[12];
t = Map[DivisorSigma[0, #] &, s];
h = Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]];
Reap[Do[If[# >= Product[Prime[j], {j, PrimeNu[#]}]^2, Sow[#]] &[ h[[i]] ],
  {i, Length[h]}] ][[-1, 1]]

A376834 Numbers k that have at least 1 powerful number m such that 1 < m <= k that are not prime powers such that rad(m) | k, where rad = A007947.

Original entry on oeis.org

36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 100, 102, 108, 110, 114, 120, 126, 130, 132, 138, 140, 144, 150, 156, 160, 162, 168, 170, 174, 180, 186, 190, 192, 196, 198, 200, 204, 210, 216, 220, 222, 224, 225, 228, 230, 234, 238, 240, 246, 250, 252, 255, 258, 260

Offset: 1

Michael De Vlieger, Oct 06 2024

nonn

Numbers k such that A286708 and row k of A162306 meet.

Contains A286708, since for k in A286708, m = k is such that rad(m) | k.

			Table showing select values of a(n):
   n    a(n)                A286708 Intersect row a(n) of A162306.
  ---------------------------------------------------------------
   1    36 = 2^2 * 3^2      {36}
   2    42 = 2 * 3 * 7      {36}
   3    48 = 2^4 * 3        {36}
   4    54 = 2 * 3^3        {36}
   5    60 = 2^2 * 3 * 5    {36}
   6    66 = 2 * 3 * 11     {36}
   7    72 = 2^3 * 3^2      {36, 72}
   8    78 = 2 * 3 * 13     {36, 72}
   9    84 = 2^2 * 3 * 7    {36, 72}
  14   108 = 2^2 * 3^3      {36, 72, 108}
  17   120 = 2^3 * 3 * 5    {36, 72, 100, 108}
  24   150 = 2 * 3 * 5^2    {36, 72, 100, 108, 144}

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Plot p^e | a(n) at (x,y) = (n,pi(p)), n = 1..1500, pi = A000720, with a color function indicating exponent e = 1 in black, e = 2 = red, e = 3 = orange, ..., maximum e in magenta. The indicator bar below the image represents squarefree a(n) in green, a(n) in A332785 in blue, and a(n) in A286708 in violet.

Cf. A001694, A007947, A162306, A286708.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Reap[Do[m = 1; k = 0;
  While[Nor[k == 2, m == n + 1],
    If[And[Divisible[n, #], Divisible[m, #^2], ! PrimePowerQ[m] ] &[
      rad[m]], k++]; m++];
    If[k == 2, Sow[n]], {n, 2^10}] ][[-1, 1]]

A377193 Lexicographically earliest infinite sequence of distinct positive integers such that any term j = a(n-1) with primorial kernel is followed by a prime, whereas any other term is followed by a number with prime factors p < q = Gpf(j) which do not divide j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 16, 13, 10, 27, 32, 17, 12, 19, 14, 15, 64, 23, 18, 29, 20, 81, 128, 31, 21, 25, 24, 37, 22, 35, 36, 41, 26, 33, 28, 45, 256, 43, 30, 47, 34, 39, 40, 243, 512, 53, 38, 49, 48, 59, 42, 125, 54, 61, 44, 63, 50, 729, 1024, 67, 46, 51

Offset: 1

David James Sycamore, and Michael De Vlieger Oct 19 2024

nonn

Following j = a(n-1), a term in A005932, a(n) is the smallest prime not already listed. Otherwise a(n) = smallest novel product of powers of non divisor primes of j; a number of the form: Product_{i = 0..k} p_i^e_i; p_i a prime < q = Gpf(j) which does not divide j, e_i >= 0, k = the number of primes p_i < q which do not divide j.
Adjacent terms are coprime and the greedy algorithm implied by the definition forces naked prime p to appear in advance of any multiple m*p of p; m >1.
Prime powers enter the sequence early, consequent to j having a single non divisor prime. A power of 3 is always followed by a power of 2.
Conjectures:
(i) A permutation of the positive integers in which the primes appear in order.
(ii)The sequence obeys Selcoe's theorem (see A280864) regarding numbers that have the same squarefree kernel, namely: Construct a sequence S_r = { m*r : rad(m) | r } = { k  : rad(k) = r }, squarefree r. Terms w in S_r appear in this sequence in order. This is to say, for example, that for r = 6, terms in A033845 = {6, 12, 18, 24, 36, 48, 54, ...} appear in order.

			a(1) = 1 implies a(2) = 2 since A007947(1) = A002110(1) = 1, and 2 is the earliest unrecorded prime so far, and likewise a(3) = 3. Since rad(3) = 3 is not a primorial number a(4) = 2^2 = 4, the smallest novel number derived from 2, the only non divisor prime of 3 and < 3.
a(8) = 8 implies a(9) = 11 because 8 is a term in A055932. The non divisor primes of 11 and < 11 are 2,3,5,7  and the smallest number which can be composed using some or all of these primes is a(10) = 3^2 = 9 (since 2,3,4,5,6,7,8 have all occurred previously). Consequently a(11) = 2^4 = 16, the smallest novel power of 2.
a(195) = 154 = 2*7*11, the non divisor primes < 11 are 3 and 5, so a(196) = 405 = 3^4*5 since all smaller candidates (3,5,9,15,25,45,75,81,125,135,243,375) have already appeared.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Mathematica program.
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..25000, showing primes in red, perfect prime powers in gold, squarefree composites in green, numbers in A332785 in blue, and A286708 in purple.
Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n, pi(p)), n = 1..2048, 4X vertical exaggeration, with a color function showing exponent m = 1 in black, m = 2 in red, ..., m = 11 in magenta. The color index at bottom uses the color key described immediately above.

Cf. A000027, A000040, A000961, A002110, A006530, A007947, A033845, A055932, A083720, A246655, A280864.

A378689 a(n) = product of divisors d of n that are not coreful.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 24, 1, 14, 15, 1, 1, 54, 1, 40, 21, 22, 1, 192, 1, 26, 1, 56, 1, 27000, 1, 1, 33, 34, 35, 216, 1, 38, 39, 320, 1, 74088, 1, 88, 135, 46, 1, 3072, 1, 250, 51, 104, 1, 1458, 55, 448, 57, 58, 1, 25920000, 1, 62, 189, 1, 65, 287496

Offset: 1

Michael De Vlieger, Feb 05 2025

			Table of n, a(n), and divisors that are not coreful that produce a(n) for select n:
   n     a(n)
  -----------------------------
   1       1   (empty product)
   2       1 = 1
   3       1 = 1
   4       1 = 1
   5       1 = 1
   6       6 = 1*2*3
  10      10 = 1*2*5
  12      24 = 1*2*3*4
  14      14 = 1*2*7
  15      15 = 1*3*5
  18      54 = 1*2*3*9
  20      40 = 1*2*4*5
  21      21 = 1*3*7
  22      22 = 1*2*11
  24     192 = 1*2*3*4*8
  30   27000 = 1*2*3*5*6*10*15
  36     216 = 1*2*3*4*9

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of log_10(a(n)) for n = 1..2^20.
Michael De Vlieger, Log log scatterplot of log_10(a(n)) for n = 1..2^16 (ignoring a(n) = 1, i.e., n that is a power of a prime), showing a(n) such that n is in A286708 in purple, n in A332785 in blue, n in A120944 in green, highlighting n in A002110 in large green points.

Cf. A007955, A027750, A308135 (sums), A308360 (product of coreful divisors of n).

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Times @@ Select[Divisors[n], rad[#] != r &], {n, 120}]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = my(d=divisors(n), c=rad(n), p=1); for (i=1, #d~, if (rad(d[i]) != c, p *= d[i])); p; \\ Michel Marcus, Feb 07 2025

a(n) = A007955(n) / A308360(n).

a(n) = 1 for powers of primes n (i.e., n in A000961), since d | n such that d > 1 are coreful.

cp's OEIS Frontend

A372404 Powerful k that are not prime powers such that k/rad(k) is nonsquarefree, where rad = A007947.

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A375934 Numbers whose prime factorization has a second-largest exponent that equals 1.

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A379545 Triangle read by rows where row n lists powerful divisors d | n (i.e., d in A001694).

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A379753 Numbers that set records in A379752.

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A386224 Nonsquarefree weak numbers k that are not products of primorials, whose squarefree kernel is a primorial.

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A343294 a(n+1) is the smallest preimage k such that A008477(k) = a(n) with a(1) = 100.

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A367511 Highly composite numbers h(k) = A002182(k) such that h >= rad(h)^2, where rad() = A007947().

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A376834 Numbers k that have at least 1 powerful number m such that 1 < m <= k that are not prime powers such that rad(m) | k, where rad = A007947.

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