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

0, 2, 6, 3, 12, 3, 126, 153, 765, 1050, 5348, 14850, 85050, 501200, 91280, 3661983, 25633881, 66271296, 215467945, 254861640, 5311480020, 75327142968, 122703152000, 2187956957004, 3449084839200, 19305922856220, 11327171375520, 58038845751810, 2222926571960640

Offset: 1

Author

Robert G. Wilson v, Jul 29 2018

Keywords

• nonn

Examples

			a(1) =   0 since the first semiprime is    4 =    2 *    2;
a(2) =   2 since 4*6               =      24 =    4 *    6;
a(3) =   6 since 4*6*9             =     216 =   12 *   18;
a(4) =   3 since 4*6*9*10          =    2160 =   45 *   48;
a(5) =  12 since 4*6*9*10*14       =   30240 =  168 *  180;
a(6) =   3 since 4*6*9*10*14*15    =  453600 =  172 *  175;
a(7) = 126 since 4*6*9*10*14*15*21 = 9525600 = 3024 * 3150; etc.
		

Crossrefs

Inspired by A003681, and analogous to A061057 and A061060.

Cf. A112141.

Programs

Mathematica

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]