cp's OEIS Frontend

A114412 Records in semiprime gaps ordered by merit.

%I A114412 #22 Feb 16 2025 08:32:59
%S A114412 2,3,4,6,11,19,24,28,30,32,38,47,54,70,74,107,110,112,120,126,146
%N A114412 Records in semiprime gaps ordered by merit.
%C A114412 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
%H A114412 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Semiprime.html">Semiprime.</a>
%H A114412 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeGaps.html">Prime Gaps.</a>
%F A114412 a(n) = records in A065516(n)/log_10(A001358(n)) = records in (A001358(n+1) - A001358(n))/log_10(A001358(n)).
%e A114412 Records defined in terms of A065516 and A001358:
%e A114412 .
%e A114412   n  A065516(n)  A065516(n)/log_10(A001358(n))
%e A114412   =  ==========  ==============================
%e A114412   1       2      2 / log_10(4)  = 3.32192809...
%e A114412   2       3      3 / log_10(6)  = 3.85529162...
%e A114412   3       1      1 / log_10(9)  = 1.04795163...
%e A114412   4       4      4 / log_10(10) = 4.00000000
%e A114412   5       1      1 / log_10(14) = 0.87250286...
%e A114412   6       6      6 / log_10(15) = 5.10164492...
%e A114412   7       1      1 / log_10(21) = 0.75630419...
%e A114412   8       3      3 / log_10(22) = 2.23476557...
%e A114412   9       1      1 / log_10(25) = 0.71533827...
%t A114412 sp = 4; m0 = 0;  l = {}; lim = 1000000;
%t A114412 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 *)
%Y A114412 Cf. A001358, A065516, A111870, A111871, A113688-A113693, A114403-A114411, A114412-A114422.
%K A114412 nonn,more
%O A114412 1,1
%A A114412 _Jonathan Vos Post_, Nov 25 2005
%E A114412 Corrected and extended by _Charles R Greathouse IV_, Oct 05 2006
%E A114412 a(16)-a(21) from _Donovan Johnson_, Feb 17 2010