A379154 Prime numbers p such that the interval from p to the next prime number contains a unique perfect power.

3, 13, 47, 61, 79, 97, 127, 139, 167, 193, 211, 223, 241, 251, 283, 317, 337, 359, 397, 439, 479, 509, 523, 571, 619, 673, 727, 773, 839, 887, 953, 997, 1021, 1087, 1153, 1223, 1291, 1327, 1367, 1439, 1511, 1597, 1669, 1723, 1759, 1847, 1933, 2017, 2039, 2113

Offset: 1

Author

Gus Wiseman, Dec 18 2024

Keywords

• nonn

Comments

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

Examples

			The prime after 13 is 17, and the interval (13,14,15,16,17) contains only one perfect power 16, so 13 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

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

For zero instead of one perfect power we have the primes indexed by A377436.

The indices of these primes are A377434.

Swapping "prime" with "perfect power" gives A378355, indices A378368.

For previous instead of next prime we have A378364.

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.

A116086 gives perfect powers with no primes between them and the next perfect power.

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

A377468 gives the least perfect power > n.

Cf. A023055, A045542, A052410, A069623, A376560, A377283, A377287, A377466, A378374.

Programs

Maple

N:= 10^4: # to get all entries <= N
S:={seq(seq(a^b, b = 2 .. floor(log[a](N))), a = 2 .. floor(sqrt(N)))}:
S:= sort(convert(S,list)):
J:= select(i -> nextprime(S[i]) < S[i+1] and prevprime(S[i]) > S[i-1], [$2..nops(S)-1]):
J:= [1,op(J)]:
map(prevprime, S[J]); # Robert Israel, Jan 19 2025

Mathematica

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

PARI

is_a379154(n) = isprime(n) && #select(x->ispower(x), [n+1..nextprime(n+1)-1])==1 \\ Hugo Pfoertner, Dec 19 2024

Formula

a(n) = A151799(A378364(n+1)).