A378364 Prime numbers such that the interval from the previous prime number contains a unique perfect power.

2, 5, 17, 53, 67, 83, 101, 131, 149, 173, 197, 223, 227, 251, 257, 293, 331, 347, 367, 401, 443, 487, 521, 541, 577, 631, 677, 733, 787, 853, 907, 967, 1009, 1031, 1091, 1163, 1229, 1297, 1361, 1373, 1447, 1523, 1601, 1693, 1733, 1777, 1861, 1949, 2027, 2053

Offset: 1

Author

Gus Wiseman, Dec 16 2024

Keywords

• nonn

Comments

Perfect-powers (A001597) are 1 and numbers with a proper integer root.

Examples

			The prime before 17 is 13, and the interval (13,14,15,16,17) contains only one perfect power 16, so 17 is in the sequence.
The prime before 29 is 23, and the interval (23,24,25,26,27,28,29) contains two perfect powers 25 and 27, so 29 is not in the sequence.
		

Crossrefs

For non prime powers we have A006512.

For zero instead of one perfect power we have the prime terms of A345531.

The indices of these primes are the positions of 1 in A377432.

The indices of these primes are 1 + A377434(n-1).

For more than one perfect power see A377466.

Swapping "prime" with "perfect power" gives A378374.

For next instead of previous prime we have A379154.

A000040 lists the primes, differences A001223.

A001597 lists the perfect powers, differences A053289.

A007916 lists the non perfect powers, differences A375706.

A081676 gives the greatest perfect power <= n.

A377468 gives the least perfect power > n.

Cf. A023055, A045542, A052410, A053607, A069623, A376560, A377287, A377436.

Programs

Mathematica

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[1000],PrimeQ[#]&&Length[Select[Range[NextPrime[#,-1],#],perpowQ]]==1&]