

A215231


Increasing gaps between semiprimes.


6



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
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OFFSET

1,1


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 CramÃ©r 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


LINKS

Table of n, a(n) for n=1..38.


EXAMPLE

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


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]]


PROG

(Haskell)
a215231 n = a215231_list !! (n1)
(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


CROSSREFS

Cf. A001358 (semiprimes), A131109, A215232, A217851.
Cf. A005250 (increasing gaps between primes).
Cf. A239673 (increasing gaps between sphenic numbers).
Sequence in context: A130690 A308189 A074885 * A301512 A091336 A002235
Adjacent sequences: A215228 A215229 A215230 * A215232 A215233 A215234


KEYWORD

nonn,hard,more


AUTHOR

T. D. Noe, Aug 07 2012


EXTENSIONS

a(27)a(31) from Donovan Johnson, Aug 07 2012
a(32)a(38) from Donovan Johnson, Sep 20 2012


STATUS

approved



