A364710 -id:A364710 - cp's OEIS frontend

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

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.

A368508 Powers of superprimorials S(k)^m such that both k > 1 and m > 1, where S(n) = A006939(n).

Original entry on oeis.org

144, 1728, 20736, 129600, 248832, 2985984, 35831808, 46656000, 429981696, 5159780352, 5715360000, 16796160000, 61917364224, 743008370688, 6046617600000, 8916100448256, 106993205379072, 432081216000000, 1283918464548864, 2176782336000000, 15407021574586368, 30497732496000000

Offset: 1

Michael De Vlieger, Dec 28 2023

Proper subset of A364930, which is the intersection of A286708 and A025487, and is in turn a proper subset of A364710. This is to say, a(n) is a product of primorials and is squareful and neither squarefree nor a prime power.

Michael De Vlieger, Table of n, a(n) for n = 1..1943

Cf. A002110 (squarefree kernels), A006939, A025487, A126706, A286708, A364930, A368507.

nn = 2^120; k = 2; P = 6; Q = 2 P; Union@ Reap[While[j = 2; While[Q^j < nn, Sow[Q^j]; j++]; j > 2, k++; P *= Prime[k]; Q *= P] ][[-1, 1]]

A363814 Intersection of A126706 and A055932.

Original entry on oeis.org

12, 18, 24, 36, 48, 54, 60, 72, 90, 96, 108, 120, 144, 150, 162, 180, 192, 216, 240, 270, 288, 300, 324, 360, 384, 420, 432, 450, 480, 486, 540, 576, 600, 630, 648, 720, 750, 768, 810, 840, 864, 900, 960, 972, 1050, 1080, 1152, 1200, 1260, 1296, 1350, 1440, 1458

Offset: 1

Michael De Vlieger, Dec 18 2023

Products m*P(i) of primorials P(i) = A002110(i) such that rad(m) | P(i), i > 1, m > 1, where rad(m) = A007947(m).

			Sequence contains terms k > 1 in {6 * A003586} since all are divisible by P(2) = 6 and by no prime q that does not divide 6. Therefore 12, 18, 24, etc. are in the sequence.
Sequence does not contain k > 1 in {10 * A003592} since such k are divisible by 5 but not 3. Hence, 20, 40, etc. are not in this sequence.
Sequence does not contain k > 1 in {15 * A003593} since such k are odd. Hence, 45, 135, etc. are not in this sequence, etc.

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

Cf. A002110, A007947, A013929, A024619, A055932, A056808, A126706, A364710.

Cf. A003586, A003592, A003593.

Select[Range[12, 1080, 2], And[AnyTrue[#2, # > 1 &], Length[#1] > 1, Union@ Differences@ PrimePi[#1] == {1}] & @@ Transpose@ FactorInteger[#] &]

Union of A056808 and A364710. - Michael De Vlieger, Jan 31 2024

A368507 Powers of superprimorials.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 32, 64, 128, 144, 256, 360, 512, 1024, 1728, 2048, 4096, 8192, 16384, 20736, 32768, 65536, 75600, 129600, 131072, 248832, 262144, 524288, 1048576, 2097152, 2985984, 4194304, 8388608, 16777216, 33554432, 35831808, 46656000, 67108864, 134217728

Offset: 1

Michael De Vlieger, Dec 28 2023

Numbers k = Product_{i=1..j} p_i^e_(m*(j-i+1)) for m >= 0 and j >= 1.
Let b(n) = A006939(n) and let P(n) = A002110(n).
This sequence contains {1}, A000079, A006939, certain k in A364710 (intersection of A126706 and A025487), and certain m in A364930 (intersection of A286708 and A025487).
The only prime in this sequence is 2.
Prime powers in this sequence are powers of 2.
Outside of {1, 2}, superprimorials are in A364710.
Squareful numbers in this sequence contain {2^k, k > 1}, which are in A000079, a proper subset of A246547, and {b(k)^m, k > 1, m > 1}, which are in A364930, a proper subset of A286708.

			Powers of 2 are in the sequence since 2 = P(1).
Powers of 12 are terms, since 12 = P(1)*P(2).
Powers of 360 are terms, since 360 = P(1)*P(2)*P(3), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..5301

Cf. A000079, A002110, A006939, A025487, A126706, A246547, A286708, A364710, A364930, A368508.

nn = 2^60; k = 1; P = 2; Q = 2; {1}~Join~Union@ Reap[While[j = 1; While[Q^j < nn, Sow[Q^j]; j++]; j > 1, k++; P *= Prime[k]; Q *= P] ][[-1, 1]]

A380033 Numbers that set records in A380032.

Original entry on oeis.org

12, 36, 144, 576, 720, 900, 2880, 3600, 14400, 32400, 44100, 57600, 129600, 176400, 705600, 1587600, 2822400, 6350400, 11289600, 21344400, 25401600, 57153600, 85377600, 101606400, 192099600, 341510400, 768398400, 1366041600, 3073593600, 6915585600, 12294374400

Offset: 1

Michael De Vlieger, Jan 11 2025

nonn

Proper subset of A364710 (intersection of A025487 and A126706).
Conjecture 1: Almost all numbers in this sequence are powerful squares. Only 12, 720, and 2880 are not powerful. Thereby this sequence is a proper subset of A368682 (intersection of A025487 and A131605, the latter a subset of A001597 and A286708), in turn a subset of A364710.
Conjecture 2: 36, 900, and 44100 are the only squares of primorials (in A061742) in the sequence.

			Let b(n) = A380032(n).
Table showing exponents of prime power factors of a(n) for n = 1..12.
Example: a(5) = 2880 = 2^6 * 3^2 * 5, hence we write "6.2.1".
   n     a(n)  Exp.   b(a(n))
  --------------------------
   1      12   2.1        1   2*6
   2      36   2.2        2   2*18 = 3*12
   3     144   4.2        3   2*72 = 3*48 = 4*36
   4     576   6.2        4   2*288 = 3*192 = 4*144 = 8*72
   5     720   4.2.1      5   2*360 = 3*240 = 4*180 = 6*120 = 12*60
   6     900   2.2.2      6
   7    2880   6.2.1      7
   8    3600   4.2.2      9
   9   14400   6.2.2     12
  10   32400   4.4.2     13
  11   44100   2.2.2.2   14
  12   57600   8.2.2     15

Michael De Vlieger, Table of n, a(n) for n = 1..66
Michael De Vlieger, Prime power decomposition of a(n) n = 1..66, also including n = 67..141 (asterisked) that would follow if Conjecture 1 is true.
Michael De Vlieger, List of (d, k/d), d < k/d, k = a(n), n = 1..24, d | k, d < k/d, such that gcd(d, k/d) > 1 and d | k/d but rad(k/d) does not divide d.

Cf. A025487, A061742, A126706, A364710, A380032, A380034.

(* Load function f at A025487 *)
r = 0;
s = Select[Union@ Flatten@ f[8][[3 ;; -1]], Not @* SquareFreeQ];
nn = Length[s]; Print[nn];
Reap[Monitor[
  Do[k = s[[i]];
    If[# > r, r = #; Sow[k]] &@
      Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
        _?((m = GCD @@ {##};
          And[! MemberQ[{1, #2}, m],
          m == #1,
          ! Divisible[#1, rad[#2]]]) & @@ # &)], {i, nn}], i] ][[-1, 1]]

A368325 Union of primorials and powers of 2.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 30, 32, 64, 128, 210, 256, 512, 1024, 2048, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 131072, 262144, 510510, 524288, 1048576, 2097152, 4194304, 8388608, 9699690, 16777216, 33554432, 67108864, 134217728, 223092870, 268435456, 536870912, 1073741824

Offset: 1

Michael De Vlieger, Jan 30 2024

Numbers that (1) are either prime powers or squarefree and (2) have a primorial kernel.

Michael De Vlieger, Table of n, a(n) for n = 1..3671

Cf. A000079, A000961, A002110, A005117, A025487, A055932, A126706, A303554, A363814, A364710.

Union@ Flatten@ {#, 2^Range[Log2[Last[#] ] ] } &@
  FoldList[Times, 1, Prime@ Range[nn]]

Union of A000079 and A002110.
Intersection of A055932 and A303554.
Intersection of A025487 and A303554.
{a(n)} = A055932 \ A126706 = A025487 \ A126706.
A055932 = union of this sequence and A363814.
A025487 = union of this sequence and A364710.

cp's OEIS Frontend

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

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A368508 Powers of superprimorials S(k)^m such that both k > 1 and m > 1, where S(n) = A006939(n).

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A363814 Intersection of A126706 and A055932.

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A368507 Powers of superprimorials.

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A380033 Numbers that set records in A380032.

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A368325 Union of primorials and powers of 2.

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