A304250 - OEIS

%I #28 Dec 31 2024 06:29:38

%S 4,8,16,32,36,64,128,144,216,256,324,512,576,900,1024,1296,1728,2048,

%T 2304,2916,3600,4096,5184,5832,7776,8100,8192,9216,11664,13824,14400,

%U 16384,20736,22500,26244,27000,32400,32768,36864,44100,46656,57600,65536,72900

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

%C 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 Michael De Vlieger, <a href="/A304250/b304250.txt">Table of n, a(n) for n = 1..10000</a>

%F Intersection of A001597 and A055932.

%e Sequence of all normal periodic multisets begins

%e 4:    {1,1}

%e 8:    {1,1,1}

%e 16:   {1,1,1,1}

%e 32:   {1,1,1,1,1}

%e 36:   {1,1,2,2}

%e 64:   {1,1,1,1,1,1}

%e 128:  {1,1,1,1,1,1,1}

%e 144:  {1,1,1,1,2,2}

%e 216:  {1,1,1,2,2,2}

%e 256:  {1,1,1,1,1,1,1,1}

%e 324:  {1,1,2,2,2,2}

%e 512:  {1,1,1,1,1,1,1,1,1}

%e 576:  {1,1,1,1,1,1,2,2}

%e 900:  {1,1,2,2,3,3}

%e 1024: {1,1,1,1,1,1,1,1,1,1}

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

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

%K nonn,changed

%O 1,1

%A _Gus Wiseman_, May 13 2018