A368682 -id:A368682 - cp's OEIS frontend

A304250 Perfect powers whose prime factors span an initial interval of prime numbers.

4, 8, 16, 32, 36, 64, 128, 144, 216, 256, 324, 512, 576, 900, 1024, 1296, 1728, 2048, 2304, 2916, 3600, 4096, 5184, 5832, 7776, 8100, 8192, 9216, 11664, 13824, 14400, 16384, 20736, 22500, 26244, 27000, 32400, 32768, 36864, 44100, 46656, 57600, 65536, 72900

Offset: 1

Gus Wiseman, May 13 2018

nonn

The multiset of prime indices of a(n) is the a(n)-th row of A112798. This multiset is normal, meaning it spans an initial interval of positive integers, and periodic, meaning its multiplicities have a common divisor greater than 1.

			Sequence of all normal periodic multisets begins
4:    {1,1}
8:    {1,1,1}
16:   {1,1,1,1}
32:   {1,1,1,1,1}
36:   {1,1,2,2}
64:   {1,1,1,1,1,1}
128:  {1,1,1,1,1,1,1}
144:  {1,1,1,1,2,2}
216:  {1,1,1,2,2,2}
256:  {1,1,1,1,1,1,1,1}
324:  {1,1,2,2,2,2}
512:  {1,1,1,1,1,1,1,1,1}
576:  {1,1,1,1,1,1,2,2}
900:  {1,1,2,2,3,3}
1024: {1,1,1,1,1,1,1,1,1,1}

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

Cf. A000837, A000961 A001597, A007916, A018783, A052409, A052410, A055932, A056239, A112798, A178472, A303431, A303709, A304450, A368682.

Select[Range[1000],FactorInteger[#][[-1,1]]==Prime[Length[FactorInteger[#]]]&&GCD@@FactorInteger[#][[All,2]]>1&]

Intersection of A001597 and A055932.

A380446 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k), where omega = A001221.

Original entry on oeis.org

36, 144, 216, 324, 576, 900, 1296, 1728, 2304, 2916, 3600, 5184, 5832, 7776, 8100, 9216, 11664, 13824, 14400, 20736, 22500, 26244, 27000, 32400, 36864, 44100, 46656, 57600, 72900, 82944, 90000, 104976, 110592, 129600, 147456, 157464, 176400, 186624, 202500, 216000

Offset: 1

Michael De Vlieger, Jul 25 2025

nonn

Perfect powers k^m, m > 1, for k in A055932.
Union of {k^m : rad(k) | P(i), m >= 2}, rad = A007947, P = A002110. Therefore perfect powers in A033845, A143207, A147571, A147572, etc. are proper subsets.
Terms are even. For a(n) such that omega(a(n)) > 2, a(n) mod 10 = 0, where omega = A001221.

			Table of n, a(n) for select n, showing exponents m of prime power factors p^m | a(n) for primes p listed in the heading. Terms that also appear in A368682 are marked by "#":
                         Exponents
 n      a(n)             2.3.5.7.11
-----------------------------------
 1       36 =    6^2  #  2.2
 2      144 =   12^2  #  4.2
 3      216 =    6^3  #  3.3
 4      324 =   18^2     2.4
 5      576 =   24^2  #  6.2
 6      900 =   30^2  #  2.2.2
 7     1296 =    6^4  #  4.4
 8     1728 =   12^3  #  6.3
 9     2304 =   48^2  #  8.2
10     2916 =   54^2     2.6
11     3600 =   60^2  #  4.2.2
12     5184 =   72^2  #  6.4
26    44100 =  210^2  #  2.2.2.2
90  5336100 = 2310^2  #  2.2.2.2.2

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Fast Mathematica algorithm for A055932.

Cf. A001597, A002110, A006530, A007947, A033845, A053669, A055932, A126706, A131605, A143207, A147571, A147572, A246547, A304250, A365308 (subset), A368682 (subset), A369374 (superset).

(* Load linked Mathematica algorithm, then: *)
Select[Union@ Flatten[a055932[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[All, -1]] > 1] & @@ {#, FactorInteger[#]} &]

Intersection of A131605 and A055932 = A304250 \ A246547.

A368681 Products of primorials that are perfect powers.

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 64, 128, 144, 216, 256, 512, 576, 900, 1024, 1296, 1728, 2048, 2304, 3600, 4096, 5184, 7776, 8192, 9216, 13824, 14400, 16384, 20736, 27000, 32400, 32768, 36864, 44100, 46656, 57600, 65536, 82944, 110592, 129600, 131072, 147456, 176400, 186624

Offset: 1

Michael De Vlieger, Jan 02 2024

nonn

Intersection of A025487 and A001597.
Contains A365308 (perfect powers of composite primorials), A368508 (perfect powers of composite superprimorials), and A368682.
These numbers are perfect powers of some smaller product of primorials.

			Let b(n) = A025487(n).
a(1) = b(1) = 1 = 1^k = b(1)^k, k >= 2,
a(2) = b(3) = 4 = 2^2 = b(2)^2,
a(3) = b(5) = 8 = 2^3 = b(2)^3,
a(6) = b(11) = 36 = 6^2 = b(4)^2,
a(9) = b(19) = 144 = 12^2 = b(6)^2, etc.
2 is not in the sequence since 2 is squarefree and not in A001597.

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

Cf. A000079, A001597, A002110, A025487, A131605, A365308, A368508, A368682.

{1}~Join~Select[Range[4, 200000, 2], Or[PrimePowerQ[#], And[Union@ Differences@ PrimePi@ #1 == {1}, AllTrue[Union@ Differences@ #2, # <= 0 &], GCD @@ #2 > 1] & @@ Transpose@ FactorInteger[#]] &]

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]]

A380452 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k) that are not also products of primorials, where omega = A001221.

Original entry on oeis.org

324, 2916, 5832, 8100, 11664, 22500, 26244, 72900, 90000, 104976, 157464, 202500, 236196, 291600, 360000, 396900, 419904, 562500, 656100, 729000, 944784, 1102500, 1259712, 1440000, 1822500, 1889568, 2125764, 2160900, 2250000, 2624400, 3375000, 3572100, 3779136

Offset: 1

Michael De Vlieger, Jul 25 2025

nonn

Perfect powers k^m, m > 1, for composite k in A056808.

Terms are even. For a(n) such that omega(a(n)) > 2, a(n) mod 10 = 0, where omega = A001221.

			Table of n, a(n) for select n, showing exponents m of prime power factors p^m | a(n) for primes p listed in the heading:
                      Exponents
 n      a(n)          2.3.5
-------------------------------
 1      324 =  18^2   2.4
 2     2916 =  54^2   2.6
 3     5832 =  18^3   3.6
 4     8100 =  90^2   2.4.2
 5    11664 = 108^2   4.6
 6    22500 = 150^2   2.2.4
 7    26244 = 162^2   2.8
 8    72900 = 270^2   2.6.2
 9    90000 = 300^2   4.2.4
10   104976 =  18^4   4.8
11   157464 =  54^3   3.9
12   202500 = 450^2   2.4.4

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Fast Mathematica algorithm for A055932.

Cf. A001597, A002110, A006530, A007947, A053669, A055932, A131605, A246547, A304250, A368682, A369374, A380446.

(* Load linked Mathematica algorithm, then: *)
Select[Union@ Flatten[a055932[7][[3 ;; -1, 2 ;; -1]] ], And[Divisible[#1, Apply[Times, #2[[All, 1]] ]^2], GCD @@ #2[[All, -1]] > 1] & @@ {#, FactorInteger[#]} &]

Intersection of A131605 and A056808 = A380446 \ A368682.

Set difference A380446 \ A025487.

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A304250 Perfect powers whose prime factors span an initial interval of prime numbers.

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A380446 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k), where omega = A001221.

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A368681 Products of primorials that are perfect powers.

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A380033 Numbers that set records in A380032.

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A380452 Perfect powers k^m, m > 1, omega(k) > 1, such that A053669(k) > A006530(k) that are not also products of primorials, where omega = A001221.

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