A364930 -id:A364930 - cp's OEIS frontend

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

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.

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]]

A378002 Achilles numbers that are products of primorials.

Original entry on oeis.org

72, 288, 432, 864, 1152, 1800, 2592, 3456, 4608, 5400, 6912, 7200, 10368, 10800, 15552, 18432, 21600, 27648, 28800, 31104, 41472, 43200, 54000, 55296, 62208, 64800, 73728, 86400, 88200, 93312, 108000, 115200, 124416, 162000, 165888, 172800, 194400, 221184, 259200

Offset: 1

Michael De Vlieger, Nov 16 2024

Products of primorials that are powerful but not perfect powers.

			Prime power decomposition of the first 12 terms:
   a(1) =   72 = 2^3 * 3^2
   a(2) =  288 = 2^5 * 3^2
   a(3) =  432 = 2^4 * 3^3
   a(4) =  864 = 2^5 * 3^3
   a(5) = 1152 = 2^7 * 3^2
   a(6) = 1800 = 2^3 * 3^2 * 5^2
   a(7) = 2592 = 2^5 * 3^4
   a(8) = 3456 = 2^7 * 3^3
   a(9) = 4608 = 2^9 * 3^2
  a(10) = 5400 = 2^3 * 3^3 * 5^2
  a(11) = 6912 = 2^8 * 3^3
  a(12) = 7200 = 2^5 * 3^2 * 5^2

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

Cf. A001597, A001694, A002110, A007947, A052486, A025487, A286708, A364930, A377854.

(* First load function f in A025487, then: *)
Select[Rest@ Union@ Flatten@ f[14],
 And[Divisible[#, Apply[Times, #2[[All, 1]] ]^2],
   GCD @@ #2[[All, -1]] == 1] & @@ {#, FactorInteger[#]} &]

Intersection of A286708 \ A001597 and A025487.
Intersection of A052486 and A025487.
Proper subset of A364930, in turn a proper subset of A369374.
Proper subset of A377854.

A386223 Nonsquarefree weak numbers k that are products of primorials.

Original entry on oeis.org

12, 24, 48, 60, 96, 120, 180, 192, 240, 360, 384, 420, 480, 720, 768, 840, 960, 1080, 1260, 1440, 1536, 1680, 1920, 2160, 2520, 2880, 3072, 3360, 3840, 4320, 4620, 5040, 5760, 6144, 6300, 6480, 6720, 7560, 7680, 8640, 9240, 10080, 11520, 12288, 12600, 12960, 13440

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    12   2^2 * 3
 2    24   2^3 * 3
 3    48   2^4 * 3
 4    60   2^2 * 3 * 5
 5    96   2^5 * 3
 6   120   2^3 * 3 * 5
 7   180   2^2 * 3^2 * 5
 8   192   2^6 * 3
 9   240   2^4 * 3 * 5
10   360   2^3 * 3^2 * 5
11   384   2^7 * 3
12   420   2^2 * 3 * 5 * 7

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

Cf. A001694, A002110, A007947, A052485, A025487, A055932, A126706, A286708, A332785, A364930, A380543.

(* Load May 19 2018 function f at A025487, then run the following: *)
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Select[Union@ Flatten[f[6][[3 ;; -1, 2 ;; -1]] ], ! Divisible[#, rad[#]^2] &]

Subset of A380543.
Intersection of A025487 and A332785, where A332785 = A052485 \ A005117 = A126706 \ A001694.
The union of this sequence and A364930 is A126706.

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]]

A369420 Powerful numbers k that are not prime powers, such that k has a primorial kernel but is not a product of primorials.

Original entry on oeis.org

108, 324, 648, 972, 1944, 2700, 2916, 3888, 4500, 5832, 8100, 8748, 9000, 11664, 13500, 16200, 17496, 18000, 22500, 23328, 24300, 26244, 34992, 36000, 40500, 45000, 48600, 52488, 67500, 69984, 72000, 72900, 78732, 81000, 90000, 97200, 104976, 112500, 121500, 132300

Offset: 1

Michael De Vlieger, Jan 22 2024

nonn

Numbers k such that Omega(k) > omega(k) > 1, prime powers p^m | k are such that m > 1, rad(k) is a primorial, but k is not a product of primorials, where Omega = A001222 and omega = A001221.
Contains no odd numbers as a consequence of being a proper subset of A055932.
Proper subset of A369419, which is in turn a proper subset of A126706.

			36 = 2^2 * 3^2 is a product of primorials, therefore not in the sequence.
72 = 2^3 * 3^2 is not a term because it is a product of primorials.
100 = 2^2 * 5^2 is not in the sequence since it does not have a primorial kernel.
108 = 2^2 * 3*3 is in the sequence since it is not a product of primorials, but its squarefree kernel is 6, a primorial.
144 = 2^4 * 3^2 is not in the sequence since it is a product of primorials, etc.

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

Cf. A001221, A001222, A002110, A025487, A055932, A056808, A126706, A286708, A364930, A369374.

With[{nn = 2^20},
 Select[
   Select[
     Rest@ Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}],
     Not@*PrimePowerQ],
   And[EvenQ[#1],
     Union@ Differences@ PrimePi[#2[[All, 1]]] == {1}, !
     AllTrue[Differences@ #2[[All, -1]], # <= 0 &]] & @@
     {#, FactorInteger[#]} &] ]

{a(n)} = {A369374 \ A364930}.

Intersection of A056808 and A286708.

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

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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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A378002 Achilles numbers that are products of primorials.

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A386223 Nonsquarefree weak numbers k that are products of primorials.

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A368507 Powers of superprimorials.

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A369420 Powerful numbers k that are not prime powers, such that k has a primorial kernel but is not a product of primorials.

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