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%I A290965 #31 Sep 04 2017 00:38:54
%S A290965 6,12,15,18,21,24,30,35,36,42,45,48,54,55,60,63,65,66,70,72,75,77,78,
%T A290965 84,85,90,91,95,96,102,105,108,110,114,115,119,120,126,130,132,133,
%U A290965 135,138,140,143,144,147,150,154,156,161,162,165,168,170,174,175,180,182,186,187,189,190,192,195,198,203
%N A290965 Let n = p1^e1*...*pj^ej*...*pm^em be the prime factorization of n > 1, with the primes in increasing order and ej > 0. If for some j < m the sum p1^2 + ... + pj^2 > p(j+1), then n is in the sequence.
%C A290965 Sequence is a semigroup, since it is closed under multiplication, an associative operation--in fact, it is provably superclosed, i.e., a product of a term in sequence and an arbitrary number is a term in the sequence since the preexisting primes will still be in the new number.
%C A290965 Density: There are 28 terms in the sequence less than 100. Using WolframAlpha, 72% of numbers from 10^20 + 1 through 10^20 + 50 were found to be in the sequence.
%C A290965 Other facts: No primes or prime powers are in the sequence.
%C A290965 Related sequences: Some other sequences that are superclosed semigroups are the counting numbers, the numbers that are not squarefree, and the numbers with initial product in factorization greater than a later prime in the factorization. (See crossrefs.)
%e A290965 6 = 2*3 is a term since 2^2 > 3.
%e A290965 1095 = 3*5*73 is a term because 3^2 > 5.
%e A290965 10, 20, and 100 are not terms since 2^2 < 5.
%e A290965 66 = 2*3*11 and 78 = 2*3*13 are terms since 2^2 + 3^2 > 11 and 2^2 + 3^2 = 13.
%e A290965 975560 = 2^3*5*29^3 is a term since 2^2 + 5^2 = 29.
%t A290965 Select[Range@ 203, AnyTrue[Partition[FactorInteger[#][[All, 1]], 2, 1], #1^2 > #2 & @@ # &] &] (* _Michael De Vlieger_, Aug 17 2017 *)
%Y A290965 Cf. A000027, A013929, A289484.
%K A290965 nonn
%O A290965 1,1
%A A290965 _Richard Locke Peterson_, Aug 15 2017

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