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

1,1

Comments

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

Links

Table of n, a(n) for n=1..50.

Example

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.

Mathematica

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

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.

Sequence in context: A216969 A148406 A148407 * A141303 A263678 A133510

Adjacent sequences: A378361 A378362 A378363 * A378365 A378366 A378367

Keyword

nonn,new

Author

Gus Wiseman, Dec 16 2024

Status

approved