A215231 Increasing gaps between semiprimes.

2, 3, 4, 6, 7, 11, 14, 19, 20, 24, 25, 28, 30, 32, 38, 47, 54, 55, 70, 74, 76, 82, 85, 87, 88, 95, 98, 107, 110, 112, 120, 123, 126, 146, 163, 166, 171, 174

Offset: 1

Author

T. D. Noe, Aug 07 2012

Keywords

• nonn
• hard
• more

Comments

See A215232 and A217851 for the semiprimes that begin and end the gaps.

Records in A065516. - R. J. Mathar, Aug 09 2012

How long can these gaps be? In the Cramér model, with x = A215232(n), they are of length log(x)^2/log(log(x))(1 + o(1)) with probability 1. - Charles R Greathouse IV, Sep 07 2012

a(n) = A065516(A085809(n)). - Reinhard Zumkeller, Mar 23 2014

Examples

			4 is here because the difference between 10 and 14 is 4, and there is no smaller semiprimes with this property.
		

Crossrefs

Cf. A001358 (semiprimes), A131109, A215232, A217851.

Cf. A005250 (increasing gaps between primes).

Cf. A239673 (increasing gaps between sphenic numbers).

Programs

Haskell

a215231 n = a215231_list !! (n-1)
(a215231_list, a085809_list) = unzip $ (2, 1) : f 1 2 a065516_list where
   f i v (q:qs) | q > v = (q, i) : f (i + 1) q qs
                | otherwise = f (i + 1) v qs
-- Reinhard Zumkeller, Mar 23 2014

Mathematica

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; nextSemiprime[n_] := Module[{m = n + 1}, While[! SemiPrimeQ[m], m++]; m]; t = {{0, 0}}; s1 = nextSemiprime[1]; While[s1 < 10^7, s2 = nextSemiprime[s1]; d = s2 - s1; If[d > t[[-1, 1]], AppendTo[t, {d, s1}]; Print[{d, s1}]]; s1 = s2]; t = Rest[t]; Transpose[t][[1]]

Extensions

a(27)-a(31) from Donovan Johnson, Aug 07 2012

a(32)-a(38) from Donovan Johnson, Sep 20 2012