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%I #23 Sep 08 2022 08:46:25
%S 4,6,9,26,95,74,69,118,25,10,15,34,87,82,39,214,33,358,49,94,93,206,
%T 155,14,21,122,65,254,35,86,57,262,115,106,141,46,123,166,55,382,91,
%U 346,183,38,209,194,129,58,85,466,51,158,161,398,119,134,185,62,159
%N a(1) = 4. a(n) is the smallest semiprime number, which is not an earlier term, for which a(n - 1) + a(n) is a brilliant semiprime number (A078972).
%C The brilliant semiprime numbers in order of appearance are: 10, 15, 35, 121, 169, 143, 187, 143, 35, 25, 49, 121, 169, 121, 253, 247, 391, 407, 143, 187, 299, 361, 169, 35, 143, 187, 319, 289, 121, 143, ... It is observed that some numbers repeat: 35 = 9 + 26 = 25 + 10 = 14 + 21 or 143 = 74 + 69 = 118 + 25 = 49 + 94 = 21 + 122 = 86 + 57.
%e a(1) + a(2) = 4 + 6 = A001358(1) + A001358(2) = 10 = A078972(4).
%e a(2) + a(3) = 6 + 9 = A001358(2) + A001358(3) = 15 = A078972(6).
%e a(3) + a(4) = 9 + 26 = A001358(3) + A001358(10) = 35 = A078972(9).
%e a(4) + a(5) = 26 + 95 = A001358(10) + A001358(34) = 121 = A078972(11).
%t Block[{a = {4}}, Do[Block[{k = 6}, While[Nand[FreeQ[a, k], PrimeOmega[k] == 2, If[PrimeOmega[#] == 2, SameQ @@ Map[IntegerLength, FactorInteger[#][[All, 1]] ], False] &[a[[-1]] + k]], k++]; AppendTo[a, k]], {i, 58}]; a] (* _Michael De Vlieger_, Nov 06 2020 *)
%o (Magma) bs:=func<n|#Divisors(n) eq 3 or &+[d[2]: d in Factorization(n)] eq 2 and #Intseq(Factorization(n)[1][1]) eq #Intseq(Factorization(n)[2][1])>; s:=func<n|&+[d[2]: d in Factorization(n)] eq 2>; a:=[ 4 ]; for n in [2..60] do k:=2; while k in a or not s(k) or not bs(k+a[n-1]) do k:=k+1; end while; Append(~a,k); end for; a;
%Y Cf. A001358, A078972.
%K nonn,base
%O 1,1
%A _Marius A. Burtea_, Oct 26 2020
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