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

4, 8, 16, 32, 36, 64, 128, 144, 216, 256, 324, 512, 576, 900, 1024, 1296, 1728, 2048, 2304, 2916, 3600, 4096, 5184, 5832, 7776, 8100, 8192, 9216, 11664, 13824, 14400, 16384, 20736, 22500, 26244, 27000, 32400, 32768, 36864, 44100, 46656, 57600, 65536, 72900

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..44.

Formula

Intersection of A001597 and A055932.

Example

Sequence of all normal periodic multisets begins
4:    {1,1}
8:    {1,1,1}
16:   {1,1,1,1}
32:   {1,1,1,1,1}
36:   {1,1,2,2}
64:   {1,1,1,1,1,1}
128:  {1,1,1,1,1,1,1}
144:  {1,1,1,1,2,2}
216:  {1,1,1,2,2,2}
256:  {1,1,1,1,1,1,1,1}
324:  {1,1,2,2,2,2}
512:  {1,1,1,1,1,1,1,1,1}
576:  {1,1,1,1,1,1,2,2}
900:  {1,1,2,2,3,3}
1024: {1,1,1,1,1,1,1,1,1,1}

Mathematica

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

Crossrefs

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

Sequence in context: A319180 A075090 A354955 * A368681 A322793 A283052

Adjacent sequences: A304247 A304248 A304249 * A304251 A304252 A304253

Keyword

nonn

Author

Gus Wiseman, May 13 2018

Status

approved