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Nonsquare composites n whose smallest prime factor is greater than or equal to n^(2/5).
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%I #28 May 31 2024 22:05:21

%S 15,35,55,77,91,119,143,187,209,221,247,253,299,319,323,341,377,391,

%T 403,437,481,493,527,533,551,559,589,629,667,697,703,713,731,779,799,

%U 817,851,893,899,901,943,989,1003,1007,1037,1073,1081,1121,1139,1147,1159,1189

%N Nonsquare composites n whose smallest prime factor is greater than or equal to n^(2/5).

%C Referred to as "freak-show composites" by Doug Massey (see link), who noted that they are exceptionally difficult to factor by hand using trial division.

%C 4181, 4183, 4187, and 4189 are the first four consecutive terms that differ only in the last decimal digit. The next such set of four consecutive terms begins at 806621.

%C Ankit Bisain says that for sufficiently large x, there are fewer values in this sequence less than x than there are primes less than x.

%H Quora, <a href="https://www.quora.com/What-s-the-definition-of-a-freak-show-composite-number/answer/Doug-Massey">What's the definition of a freak-show composite number?</a>

%e The smallest prime factor of 4189 is 59, which is greater than 4189^(2/5).

%p q:= n-> not (isprime(n) or issqr(n) or min(numtheory[factorset](n))^5<n^2):

%p select(q, [$4..2030])[]; # _Alois P. Heinz_, Jan 07 2021

%t nscQ[n_]:=CompositeQ[n]&&!IntegerQ[Sqrt[n]]&&FactorInteger[n][[1,1]]>= Surd[n^2,5]; Select[Range[1200],nscQ] (* _Harvey P. Dale_, Jul 25 2021 *)

%o (PARI) isok(n) = (n>1) && !isprime(n) && !issquare(n) && (factor(n)[1,1]^5 >= n^2); \\ _Michel Marcus_, Jan 07 2021

%Y Superset of A006094, except for initial 6 in that sequence.

%Y Subsequence of A089229.

%K nonn

%O 1,1

%A _Phil Scovis_, Jan 07 2021