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

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

1,1

Comments

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Michael De Vlieger, Fast Mathematica algorithm for A055932.

Formula

Intersection of A131605 and A055932 = A304250 \ A246547.

Example

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

Mathematica

(* 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[#]} &]

Crossrefs

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

Sequence in context: A138202 A204106 A049227 * A368682 A016910 A005017

Adjacent sequences: A380443 A380444 A380445 * A380447 A380448 A380449

Keyword

nonn

Author

Michael De Vlieger, Jul 25 2025

Status

approved