A379157 Prime powers p such that the interval from p to the next prime power contains a unique prime number.

3, 4, 7, 9, 13, 16, 23, 27, 31, 32, 47, 49, 61, 64, 79, 81, 113, 125, 127, 128, 167, 169, 241, 243, 251, 256, 283, 289, 337, 343, 359, 361, 509, 512, 523, 529, 619, 625, 727, 729, 839, 841, 953, 961, 1021, 1024, 1327, 1331, 1367, 1369, 1669, 1681, 1847, 1849

Offset: 1

Author

Gus Wiseman, Dec 22 2024

Keywords

• nonn

Examples

			The next prime power after 32 is 37, with interval (32,33,34,35,36,37) containing just one prime 37, so 32 is in the sequence.
		

Crossrefs

For no primes we have A068315/A379156, for perfect powers A116086/A274605.

The previous instead of next prime power we have A175106.

For perfect powers instead of prime powers we have A378355.

The positions of these prime powers (in A246655) are A379155.

A000015 gives the least prime power >= n.

A000040 lists the primes, differences A001223.

A000961 lists the powers of primes, differences A057820.

A031218 gives the greatest prime power <= n.

A065514 gives the greatest prime power < prime(n), difference A377289.

A246655 lists the prime powers.

A366833 counts prime powers between primes, see A053607, A304521.

A366835 counts primes between prime powers, for perfect powers A080769.

Cf. A046933, A067871, A080101, A178700, A345531, A377281, A377287, A377434, A378374.

Programs

Mathematica

v=Select[Range[100],PrimePowerQ]
nextpripow[n_]:=NestWhile[#+1&,n+1,!PrimePowerQ[#]&]
Select[v,Length[Select[Range[#,nextpripow[#]],PrimeQ]]==1&]

Formula

a(n) = A246655(A379155(n)).