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

%I A304250 #28 Dec 31 2024 06:29:38
%S A304250 4,8,16,32,36,64,128,144,216,256,324,512,576,900,1024,1296,1728,2048,
%T A304250 2304,2916,3600,4096,5184,5832,7776,8100,8192,9216,11664,13824,14400,
%U A304250 16384,20736,22500,26244,27000,32400,32768,36864,44100,46656,57600,65536,72900
%N A304250 Perfect powers whose prime factors span an initial interval of prime numbers.
%C A304250 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.
%H A304250 Michael De Vlieger, <a href="/A304250/b304250.txt">Table of n, a(n) for n = 1..10000</a>
%F A304250 Intersection of A001597 and A055932.
%e A304250 Sequence of all normal periodic multisets begins
%e A304250 4:    {1,1}
%e A304250 8:    {1,1,1}
%e A304250 16:   {1,1,1,1}
%e A304250 32:   {1,1,1,1,1}
%e A304250 36:   {1,1,2,2}
%e A304250 64:   {1,1,1,1,1,1}
%e A304250 128:  {1,1,1,1,1,1,1}
%e A304250 144:  {1,1,1,1,2,2}
%e A304250 216:  {1,1,1,2,2,2}
%e A304250 256:  {1,1,1,1,1,1,1,1}
%e A304250 324:  {1,1,2,2,2,2}
%e A304250 512:  {1,1,1,1,1,1,1,1,1}
%e A304250 576:  {1,1,1,1,1,1,2,2}
%e A304250 900:  {1,1,2,2,3,3}
%e A304250 1024: {1,1,1,1,1,1,1,1,1,1}
%t A304250 Select[Range[1000],FactorInteger[#][[-1,1]]==Prime[Length[FactorInteger[#]]]&&GCD@@FactorInteger[#][[All,2]]>1&]
%Y A304250 Cf. A000837, A000961 A001597, A007916, A018783, A052409, A052410, A055932, A056239, A112798, A178472, A303431, A303709, A304450, A368682.
%K A304250 nonn
%O A304250 1,1
%A A304250 _Gus Wiseman_, May 13 2018