A304250 -id:A304250 - cp's OEIS frontend

A100778 Integer powers of primorial numbers.

1, 2, 4, 6, 8, 16, 30, 32, 36, 64, 128, 210, 216, 256, 512, 900, 1024, 1296, 2048, 2310, 4096, 7776, 8192, 16384, 27000, 30030, 32768, 44100, 46656, 65536, 131072, 262144, 279936, 510510, 524288, 810000, 1048576, 1679616, 2097152, 4194304, 5336100

Offset: 1

Amarnath Murthy, Nov 28 2004

Smallest squarefree numbers or their powers with distinct prime signatures. Or least numbers with prime signatures (p*q*r*...)^k, where p,q,r,... are primes and k is a whole number.
Also Heinz numbers of uniform integer partitions whose union is an initial interval of positive integers. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all uniform integer partitions whose Heinz numbers belong to the sequence begins: (1), (11), (12), (111), (1111), (123), (11111), (1122), (111111), (1111111), (1234), (111222), (11111111), (111111111), (112233), (1111111111). - Gus Wiseman, Dec 26 2018
From Amiram Eldar, Sep 26 2023: (Start)
Intersection of A025487 and A072774.
The distinct terms of A046523(A072774(n)) in ascending orders.
The k-th power of the n-th primorial number, A002110(n)^k, has (k+1)^n divisors which are the set of the (k+1)-free prime(n)-smooth numbers. (End)

			10 is not a term as 6 is a member with the same prime signature 10 > 6.
216 is a term as 216 = (2*3)^3. 243 is not a term as 32 represents that prime signature.

David A. Corneth, Table of n, a(n) for n = 1..8606 (terms <= 10^1000)

Cf. A000961, A001597, A002110, A007947, A025487, A046523, A055932, A056239, A057588, A072774, A072777, A112798, A304250, A319169, A322792, A322793.

unintQ[n_]:=And[SameQ@@Last/@FactorInteger[n],Length[FactorInteger[n]]==PrimePi[FactorInteger[n][[-1,1]]]];
Select[Range[1000],unintQ] (* Gus Wiseman, Dec 26 2018 *)

Sum_{n>=1} 1/a(n) = 1 + Sum_{n>=1} 1/A057588(n) = 2.2397359032... - Amiram Eldar, Oct 20 2020; corrected by Hal M. Switkay and Amiram Eldar, Apr 12 2021

More terms and simpler definition from Ray Chandler, Nov 29 2004

A322792 Irregular triangle read by rows where T(n,k) = A002110(n/d)^d, where d = A027750(n,k) and A002110(m) is the product of the first m primes.

Original entry on oeis.org

2, 6, 4, 30, 8, 210, 36, 16, 2310, 32, 30030, 900, 216, 64, 510510, 128, 9699690, 44100, 1296, 256, 223092870, 27000, 512, 6469693230, 5336100, 7776, 1024, 200560490130, 2048, 7420738134810, 901800900, 9261000, 810000, 46656, 4096, 304250263527210, 8192

Offset: 1

Gus Wiseman, Dec 26 2018

A reordering of A100778 (powers of primorials), these are the Heinz numbers of uniform integer partitions of length n whose union is an initial interval of positive integers. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			Triangle begins:
           2
           6          4
          30          8
         210         36         16
        2310         32
       30030        900        216         64
      510510        128
     9699690      44100       1296        256
   223092870      27000        512
  6469693230    5336100       7776       1024
Corresponding triangle of integer partitions whose Heinz numbers belong to the triangle begins:
  (1)
  (21)        (11)
  (321)       (111)
  (4321)      (2211)      (1111)
  (54321)     (11111)
  (654321)    (332211)    (222111)    (111111)
  (7654321)   (1111111)
  (87654321)  (44332211)  (22221111)  (11111111)
  (987654321) (333222111) (111111111)

First column is A002110.

Cf. A000961, A001597, A002110, A007947, A027750, A025487, A047966, A055932, A056239, A072774, A100778, A304250, A322793.

Table[Product[Prime[i]^d,{i,n/d}],{n,12},{d,Divisors[n]}]

Name edited by Peter Munn, Mar 05 2025

A322793 Proper powers of primorial numbers.

Original entry on oeis.org

4, 8, 16, 32, 36, 64, 128, 216, 256, 512, 900, 1024, 1296, 2048, 4096, 7776, 8192, 16384, 27000, 32768, 44100, 46656, 65536, 131072, 262144, 279936, 524288, 810000, 1048576, 1679616, 2097152, 4194304, 5336100, 8388608, 9261000

Offset: 1

Gus Wiseman, Dec 26 2018

nonn

A primorial number is a product of the first n primes, for some n.

Also Heinz numbers of non-strict uniform integer partitions whose union is an initial interval of positive integers. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of all non-strict uniform integer partitions whose Heinz numbers belong to the sequence begins: (11), (111), (1111), (11111), (2211), (111111), (1111111), (222111), (11111111), (111111111), (332211), (1111111111), (22221111).

Michael De Vlieger, Table of n, a(n) for n = 1..8255

Cf. A000961, A001597, A001694, A002110, A025487, A047966, A055932, A056239, A072774, A072777, A100778, A112798, A304250, A322792.

unintpropQ[n_]:=And[SameQ@@Last/@FactorInteger[n],FactorInteger[n][[1,2]]>1,Length[FactorInteger[n]]==PrimePi[FactorInteger[n][[-1,1]]]];
Select[Range[10000],unintpropQ]
(* Second program: *)
nn = 2^24; k = 1; P = 2; Union@ Reap[While[j = 2; While[P^j < nn, Sow[P^j]; j++]; j > 2, k++; P *= Prime[k]]][[-1, 1]] (* Michael De Vlieger, Oct 04 2023 *)

Sum_{n>=1} 1/a(n) = Sum_{k>=1} 1/(A002110(k)*(A002110(k)-1)) = 0.53450573145072369022... . - Amiram Eldar, Mar 10 2024

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.

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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A100778 Integer powers of primorial numbers.

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A322792 Irregular triangle read by rows where T(n,k) = A002110(n/d)^d, where d = A027750(n,k) and A002110(m) is the product of the first m primes.

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A322793 Proper powers of primorial 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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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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