A373077 Perfect powers that are sandwiched between squarefree numbers.

4, 16, 32, 36, 128, 144, 196, 216, 256, 400, 484, 900, 1156, 1296, 1600, 1728, 1764, 2048, 2704, 2916, 3136, 3364, 3600, 4356, 5184, 6084, 7056, 7396, 7744, 8100, 8192, 8464, 8836, 9216, 10404, 10816, 11236, 11664, 12100, 12544, 12996, 16384, 16900, 19044, 19600

Offset: 1

Author

Massimo Kofler, May 22 2024

Keywords

• nonn

Comments

All terms are even numbers.

Examples

			4 = 2^2 (between 3 which is a prime number and 5 which is a prime number).
16 = 2^4 (between 15 = 3 * 5 and 17 which is a prime number).
32 = 2^5 (between 31 which is a prime number and 33 = 3 * 11).
36 = 2^2 * 3^2 (between 35 = 5 * 7 and 37 which is a prime number).
		

Crossrefs

Intersection of A001597 (or A075090) and A067874.

Cf. A005117.

Programs

Maple

N:= 10^5:
S:= {}:
for n from 2 to isqrt(N) by 2 do
  for k from 2 do
    m:= n^k;
    if m > N then break fi;
    if numtheory:-issqrfree(m-1) and numtheory:-issqrfree(m+1) then S:= S union {m} fi
od od:
sort(convert(S,list)); # Robert Israel, May 22 2024

Mathematica

Select[Range[4,20000,4], GCD @@ FactorInteger[#][[;; , 2]] > 1 && And @@ SquareFreeQ /@ (# + {-1, 1}) &] (* Amiram Eldar, May 22 2024 *)

PARI

isok(k) = ispower(k) && issquarefree(k-1) && issquarefree(k+1); \\ Michel Marcus, May 22 2024