A387732 Highly powerful numbers that are perfect powers.

1, 4, 8, 16, 32, 64, 128, 144, 216, 1296, 1728, 5184, 7776, 20736, 129600, 216000, 518400, 12960000, 46656000, 116640000, 635040000, 16003008000, 280052640000, 1120210560000, 4480842240000, 21300003648000

Offset

1,2

Comments

Intersection of A001597 and A005934.

Since both A001597 and A005934 are proper subsets of A001694, numbers in A005934 that remain after subtracting those terms in this sequence are in A052486 (i.e., Achilles numbers).

Since numbers in A005934 are also in A025487, Achilles numbers in A005934 are in A378002. Terms in this sequence that are not prime powers are in A368682, while the prime powers are in A000079.

Numbers in A005934 exceeding the largest term in A307703 are in A388293.

Are there any larger terms in this sequence?

Links

Table of n, a(n) for n=1..26.

Peter Luschny, Python implementation.

Index entries for sequences related to powerful numbers.

Example

Asterisks denote perfect squares in A307703, the sequence of highly powerful numbers that are not cubefull.
                                         p-adic valuation of a(n)
 n     k     a(n) = A005934(k)           2.3.5.7.11
-----------------------------------------------------------------
 1     1                1
 2     2                4* =       2^2   2
 3     3                8  =       2^3   3
 4     4               16  =       2^4   4
 5     5               32  =       2^5   5
 6     6               64  =       2^6   6
 7     7              128  =       2^7   7
 8     8              144* =      12^2   4.2
 9     9              216  =       6^3   3.3
10    13             1296  =       6^4   4.4
11    14             1728  =      12^3   6.3
12    17             5184  =      72^2   6.4
13    18             7776  =       6^5   5.5
14    21            20736  =      12^4   8.4
15    27           129600* =     360^2   6.4.2
16    29           216000  =      60^3   6.3.3
17    33           518400* =     720^2   8.4.2
18    44         12960000  =      60^4   8.4.4
19    50         46656000  =     360^3   9.6.3
20    53        116640000  =   10800^2   8.6.4
21    62        635040000* =   25200^2   8.4.4.2
22    76      16003008000  =    2520^3   9.6.3.3
23    87     280052640000  =  529200^2   8.6.4.4
24    96    1120210560000  = 1058400^2  10.6.4.4
25   102    4480842240000  = 2116800^2  12.6.4.4
26   113   21300003648000  =   27720^3   9.6.3.3.3

Mathematica

r = -1; {1}~Join~Reap[Do[If[#2 > r, r = #2; If[#1, Sow[n] ] ] & @@ {GCD @@ # != 1, Times @@ #} &[FactorInteger[n][[;; , -1]] ], {n, 2^20}] ][[-1, 1]]

Prog

(Python) # Cf. links.

Crossrefs

Cf. A000079, A001597, A001694, A005934, A025487, A055932, A052486, A131605, A246547, A307703, A368682, A378002, A388293.

Sequence in context: A005934 A085629 A349111 * A307870 A330873 A233442

Adjacent sequences: A387729 A387730 A387731 * A387733 A387734 A387735

Keyword

nonn,more

Author

Michael De Vlieger, Oct 06 2025

Status

approved