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%I #16 Sep 08 2022 08:46:25
%S 8,10,12,13,14,15,16,18,19,20,21,23,24,25,27,28,29,30,31,34,35,36,39,
%T 40,41,42,44,45,46,49,50,53,55,56,58,59,60,63,64,70,74,84,98,125,127,
%U 130,131,135,136,142,146,147,149,152,153,156,157,158,164,168,170
%N Numbers that can be written as the sum of two brilliant numbers (A078972).
%C The sequence is infinite.
%C There are an infinite number of term pairs (a(k), a(k + 1)) that are consecutive numbers. Indeed, if p is a prime number, then 9 + p^2 and 10 + p^2 are terms. Also, numbers 14 + p^2 and 15 + p^2 are terms.
%C There are also larger sequences of consecutive numbers that are terms. For example, the 21 consecutive numbers 780, 781, ..., 800 or 4184, 4185, ..., 4204 are terms.
%e 8 = 4 + 4 = A078972(1) + A078972(1), so 8 is a term.
%e 10 = 4 + 6 = A078972(1) + A078972(2), so 10 is a term.
%e 15 = 6 + 9 = A078972(2) + A078972(3), so 15 is a term.
%t m = 200; brils = Select[Range[m], (f = FactorInteger[#])[[;; , 2]] == {2} || f[[;; , 2]] == {1, 1} && Equal @@ IntegerLength@f[[;; , 1]] &]; Select[Range[m], Length[IntegerPartitions[#, {2}, brils]] > 0 &] (* _Amiram Eldar_, Dec 06 2020 *)
%o (Magma) f:=Factorization; br:=func<n|#Divisors(n) eq 3 or &+[d[2]:d in f(n)] eq 2 and #Intseq(f(n)[1][1]) eq #Intseq(f(n)[2][1])>; [k:k in [4..200]|exists(i){m:m in [4..k-4]|br(m) and br(k-m)}];
%Y Cf. A078972, A338474.
%K nonn,base
%O 1,1
%A _Marius A. Burtea_, Dec 06 2020
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