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A373077 Perfect powers that are sandwiched between squarefree numbers.

%I A373077 #21 May 24 2024 01:38:29
%S A373077 4,16,32,36,128,144,196,216,256,400,484,900,1156,1296,1600,1728,1764,
%T A373077 2048,2704,2916,3136,3364,3600,4356,5184,6084,7056,7396,7744,8100,
%U A373077 8192,8464,8836,9216,10404,10816,11236,11664,12100,12544,12996,16384,16900,19044,19600
%N A373077 Perfect powers that are sandwiched between squarefree numbers.
%C A373077 All terms are even numbers.
%e A373077 4 = 2^2 (between 3 which is a prime number and 5 which is a prime number).
%e A373077 16 = 2^4 (between 15 = 3 * 5 and 17 which is a prime number).
%e A373077 32 = 2^5 (between 31 which is a prime number and 33 = 3 * 11).
%e A373077 36 = 2^2 * 3^2 (between 35 = 5 * 7 and 37 which is a prime number).
%p A373077 N:= 10^5:
%p A373077 S:= {}:
%p A373077 for n from 2 to isqrt(N) by 2 do
%p A373077   for k from 2 do
%p A373077     m:= n^k;
%p A373077     if m > N then break fi;
%p A373077     if numtheory:-issqrfree(m-1) and numtheory:-issqrfree(m+1) then S:= S union {m} fi
%p A373077 od od:
%p A373077 sort(convert(S,list)); # _Robert Israel_, May 22 2024
%t A373077 Select[Range[4,20000,4], GCD @@ FactorInteger[#][[;; , 2]] > 1 && And @@ SquareFreeQ /@ (# + {-1, 1}) &] (* _Amiram Eldar_, May 22 2024 *)
%o A373077 (PARI) isok(k) = ispower(k) && issquarefree(k-1) && issquarefree(k+1); \\ _Michel Marcus_, May 22 2024
%Y A373077 Intersection of A001597 (or A075090) and A067874.
%Y A373077 Cf. A005117.
%K A373077 nonn
%O A373077 1,1
%A A373077 _Massimo Kofler_, May 22 2024