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Perfect powers that are sandwiched between squarefree numbers.
1

%I #21 May 24 2024 01:38:29

%S 4,16,32,36,128,144,196,216,256,400,484,900,1156,1296,1600,1728,1764,

%T 2048,2704,2916,3136,3364,3600,4356,5184,6084,7056,7396,7744,8100,

%U 8192,8464,8836,9216,10404,10816,11236,11664,12100,12544,12996,16384,16900,19044,19600

%N Perfect powers that are sandwiched between squarefree numbers.

%C All terms are even numbers.

%e 4 = 2^2 (between 3 which is a prime number and 5 which is a prime number).

%e 16 = 2^4 (between 15 = 3 * 5 and 17 which is a prime number).

%e 32 = 2^5 (between 31 which is a prime number and 33 = 3 * 11).

%e 36 = 2^2 * 3^2 (between 35 = 5 * 7 and 37 which is a prime number).

%p N:= 10^5:

%p S:= {}:

%p for n from 2 to isqrt(N) by 2 do

%p for k from 2 do

%p m:= n^k;

%p if m > N then break fi;

%p if numtheory:-issqrfree(m-1) and numtheory:-issqrfree(m+1) then S:= S union {m} fi

%p od od:

%p sort(convert(S,list)); # _Robert Israel_, May 22 2024

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

%o (PARI) isok(k) = ispower(k) && issquarefree(k-1) && issquarefree(k+1); \\ _Michel Marcus_, May 22 2024

%Y Intersection of A001597 (or A075090) and A067874.

%Y Cf. A005117.

%K nonn

%O 1,1

%A _Massimo Kofler_, May 22 2024