A114412 Records in semiprime gaps ordered by merit.

2, 3, 4, 6, 11, 19, 24, 28, 30, 32, 38, 47, 54, 70, 74, 107, 110, 112, 120, 126, 146

Offset

1,1

Comments

There is an associated index list n = 1, 2, 4, 6, 34, 422, 1765, 4585, 8112, 8650, 8861, 75150, ... and an associated semiprime list A001358(n) = 4, 6, 10, 15, 1418, 6559, 17965, 32777, 35103, 35981, 340894, ... - R. J. Mathar, Mar 15 2009

Links

Table of n, a(n) for n=1..21.

Eric Weisstein's World of Mathematics, Semiprime.

Eric Weisstein's World of Mathematics, Prime Gaps.

Formula

a(n) = records in A065516(n)/log_10(A001358(n)) = records in (A001358(n+1) - A001358(n))/log_10(A001358(n)).

Example

Records defined in terms of A065516 and A001358:
.
  n  A065516(n)  A065516(n)/log_10(A001358(n))
  =  ==========  ==============================
  1       2      2 / log_10(4)  = 3.32192809...
  2       3      3 / log_10(6)  = 3.85529162...
  3       1      1 / log_10(9)  = 1.04795163...
  4       4      4 / log_10(10) = 4.00000000
  5       1      1 / log_10(14) = 0.87250286...
  6       6      6 / log_10(15) = 5.10164492...
  7       1      1 / log_10(21) = 0.75630419...
  8       3      3 / log_10(22) = 2.23476557...
  9       1      1 / log_10(25) = 0.71533827...

Mathematica

sp = 4; m0 = 0;  l = {}; lim = 1000000;
For[i = 5, i <= lim, i++, If[PrimeOmega[i] == 2, m = (i - sp)/Log[sp]; If[m > m0, m0 = m; AppendTo[l, i - sp]]; sp = i] ]; l (* Robert Price, Oct 29 2018 *)

Crossrefs

Cf. A001358, A065516, A111870, A111871, A113688-A113693, A114403-A114411, A114412-A114422.

Sequence in context: A111124 A295681 A117308 * A352819 A016038 A003099

Adjacent sequences: A114409 A114410 A114411 * A114413 A114414 A114415

Keyword

nonn,more

Author

Jonathan Vos Post, Nov 25 2005

Extensions

Corrected and extended by Charles R Greathouse IV, Oct 05 2006

a(16)-a(21) from Donovan Johnson, Feb 17 2010

Status

approved