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A304250
Perfect powers whose prime factors span an initial interval of prime numbers.
3
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.
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&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 13 2018
STATUS
approved