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A317490 a(n) = min {i - j} where i*j is a factorization of the n-th partial product of the semiprimes (A112141).

%I A317490 #26 Aug 11 2018 14:10:31
%S A317490 0,2,6,3,12,3,126,153,765,1050,5348,14850,85050,501200,91280,3661983,
%T A317490 25633881,66271296,215467945,254861640,5311480020,75327142968,
%U A317490 122703152000,2187956957004,3449084839200,19305922856220,11327171375520,58038845751810,2222926571960640
%N A317490 a(n) = min {i - j} where i*j is a factorization of the n-th partial product of the semiprimes (A112141).
%e A317490 a(1) =   0 since the first semiprime is    4 =    2 *    2;
%e A317490 a(2) =   2 since 4*6               =      24 =    4 *    6;
%e A317490 a(3) =   6 since 4*6*9             =     216 =   12 *   18;
%e A317490 a(4) =   3 since 4*6*9*10          =    2160 =   45 *   48;
%e A317490 a(5) =  12 since 4*6*9*10*14       =   30240 =  168 *  180;
%e A317490 a(6) =   3 since 4*6*9*10*14*15    =  453600 =  172 *  175;
%e A317490 a(7) = 126 since 4*6*9*10*14*15*21 = 9525600 = 3024 * 3150; etc.
%t A317490 SemiPrimePi[n_] := Sum[PrimePi[n/Prime[i]] - i + 1, {i, PrimePi[Sqrt[n]]}]; SemiPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[SemiPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; a[n_] := Block[{sp = Times @@ Array[SemiPrime@# &, n], d}, d = DivisorSigma[0, sp]/2; -Subtract @@  Take[ Divisors@ sp, {d, d + 1}]]; a[1] = 0; Array[a, 29]
%Y A317490 Inspired by A003681, and analogous to A061057 and A061060.
%Y A317490 Cf. A112141.
%K A317490 nonn
%O A317490 1,2
%A A317490 _Robert G. Wilson v_, Jul 29 2018