A364702 -id:A364702 - cp's OEIS frontend

A364996 Union of A360767 and A363082.

12, 18, 20, 24, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 90, 92, 99, 104, 116, 117, 120, 124, 126, 132, 136, 140, 148, 150, 152, 153, 156, 164, 168, 171, 172, 175, 176, 180, 184, 188, 198, 204, 207, 208, 212, 220, 228, 232, 234, 236, 244, 248, 260, 261

Offset: 1

Michael De Vlieger, Aug 26 2023

nonn

			This sequence is A126706 \ A361098.
Union of A364997, A364998, A364999.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, List of A126706(1..400), 20 in a row. Terms in this sequence are circled, while those in small colored circles appear in A361098. Blue represents numbers in A364702, purple A359280, and magenta A303606.
Michael De Vlieger, Plot b(n) at (x,y) = (n mod 1024, -floor(n/1024)), where terms in this sequence are shown in black, and those in A361098 appear in white.

Cf. A013929, A024619, A126706, A360767, A361098, A363082, A364997, A364998, A364999.

Select[Select[Range[261], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, Or[p r > k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

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 }.

A360589 Numbers k that set records in A355432.

Original entry on oeis.org

1, 18, 48, 54, 162, 384, 486, 1350, 1458, 2250, 2430, 3750, 6000, 6750, 7290, 11250, 12150, 14580, 15000, 15360, 18750, 21870, 30720, 33750, 36450, 37500, 43740, 56250, 61440, 65610, 93750, 122880, 168750, 182250, 187500, 196830, 245760, 281250, 328050, 360150, 375000, 393660

Offset: 1

Michael De Vlieger, Feb 22 2023

nonn

Subset of A055932.
For n > 1, subset of A360768, which is in turn a subset of A126706.
Conjecture: for n > 2, subset of A364702. - Michael De Vlieger, Oct 04 2024

			Let rad(m) = A007947(m).
a(1) = 1 since 1 is the empty product.
a(2) = 18 since {12} is a nondivisor k < 18 such that rad(k) = rad(18).
a(3) = 48 since {18, 36} are nondivisors k < 48 such that rad(k) = rad(48).
a(4) = 54 since {12, 24, 36, 48} are nondivisors k < 54 such that rad(k) = rad(54), etc.
Table shows prime decomposition of a(n) = Product p^e, noting multiplicity e in the pi(p)-th position. For example, a(n) = 1350 = 2 * 3^3 * 5^2, hence we write 1.3.2.
a(n) = A055932(i) and has A360912(n) nondivisors k < a(n) such that rad(k) = rad(a(n)).
   n    a(n) A067255(a(n))  i  A360912(n)
  ----------------------------------------
   1      1      0          1          0
   2     18      1.2        8          1
   3     48      4.1       13          2
   4     54      1.3       14          4
   5    162      1.4       25          8
   6    384      7.1       37         10
   7    486      1.5       42         14
   8   1350      1.3.2     65         16
   9   1458      1.6       67         21
  10   2250      1.2.3     81         23
  11   2430      1.5.1     85         26
  12   3750      1.1.4     99         33
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..2071
Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n, pi(p)), n = 1..2071, 24X vertical exaggeration, with a color function that represents m = 1 in black, m = 2 in red, m = 3 in orange, ... m = 34 in magenta. (Represents column "A067255(a(n))" in table in Example below.)

Cf. A007947, A055932, A126706, A355432, A360768, A360912, A364702.

rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; t = Select[Range[2^14], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]; s = Select[t, #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]; t = Table[m = s[[n]]; r = rad[m]; Count[TakeWhile[t, # < m &], _?(And[rad[#] == r, Mod[m, #] != 0] &)], {n, Length[s]}]; {1}~Join~Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]]

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]]

A367708 Numbers k that are neither squarefree nor prime powers such that max(A119288(k), A053669(k)) <= A003557(k) < A007947(k).

Original entry on oeis.org

50, 75, 80, 98, 112, 135, 147, 189, 240, 242, 245, 252, 270, 294, 300, 336, 338, 350, 352, 360, 363, 378, 396, 416, 450, 468, 480, 490, 504, 507, 525, 528, 540, 550, 560, 578, 588, 594, 600, 605, 612, 624, 650, 672, 684, 700, 702, 720, 722, 726, 735, 750, 756

Offset: 1

Michael De Vlieger, Feb 09 2024

nonn

Does not contain 3-smooth numbers.
Contains neither A168263 nor A367511.
Conjecture: contains most highly composite numbers.

			Let q = A053669(k) and let p = A119288(k).
For s = 10, we have {50, 80}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 10*{ max(5, 3) <= m < 10 : rad(m) | 10 }
   = 10*{5, 8} = {50, 80}.
For s = 15, we have {45, 135}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 15*{ max(5, 2) <= m < 15 : rad(m) | 15 }
   = 15*{5, 9} = {240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810}.
For s = 30, we have {45, 135}, since
    s * { max(p, q) <= m < s  : rad(m) | s  }
   = 30*{ max(3, 7) <= m < 30 : rad(m) | 30 }
   = 30*{8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27}
   = {240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810}.

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

Cf. A002182, A003586, A053669, A119288, A120944, A168263, A341645, A361098, A364702, A366250, A367511.

nn = 756;
Select[Select[Range[12, nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &],
  And[Max[#2, #3] <= #1 < #4, ! AllTrue[#5, # > 1 &]] & @@
    {#1/#4, #2, #3, #4, #5} & @@
    {#1, #2[[2, 1]], #3, Times @@ #2[[All, 1]], #2[[All, -1]]} & @@
    {#, FactorInteger[#], If[OddQ[#], 2,
        q = 3; While[Divisible[#, q], q = NextPrime[q]]; q]} &]

Union of {k = m*s : rad(m) | s, max(p, q) <= m < s}, where s is in A120944.
{a(n)} = A364702 \ A366250.
{a(n)} = A361098 \ A341645.

cp's OEIS Frontend

A364996 Union of A360767 and A363082.

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

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A360589 Numbers k that set records in A355432.

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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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A367708 Numbers k that are neither squarefree nor prime powers such that max(A119288(k), A053669(k)) <= A003557(k) < A007947(k).

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