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%I #19 Jan 03 2022 12:12:16
%S 9,25,39,95,267,2369,6559,8817,13705,15261,21583,35981,66921,113009,
%T 340891,783757,872219,3058853,3586843,5835191,12345473,108994623,
%U 248706917,268749691,679956119,709239621,3648864859,3790337723,4171420481,33955869693,34279038379
%N Start of record gap in odd semiprimes A046315.
%C 3 of the first 5 values of record gaps in odd semiprimes are also record merits = [A046315(k+1)-A046315(k)]/log(A046315(k)]/), namely: (15 - 9) / log(9) = 6.28770982; (111 - 95) / log(95) = 8.09010923; (287 - 267) / log(267) = 8.24228608. It is easy to prove that there are gaps of arbitrary length in even semiprimes (A100484); can we prove that there are gaps of arbitrary length in odd semiprimes (A046315) and in semiprimes (A001358)?
%C The record gaps have lengths 6, 8, 10, 16, 20, 22, 24, 26, 28, 32, 36, 38, 40, 44, 50, 52, 60, 64, 70, 74. - _T. D. Noe_, Feb 03 2006
%F {a(n)} = {A046315(k) such that A046315(k+1)-A046315(k) is a record}.
%e a(1) = A046315(2)-A046315(1) = 15 - 9 = 6.
%e a(2) = A046315(5)-A046315(4) = 33 - 25 = 8.
%e a(3) = A046315(8)-A046315(7) = 49 - 39 = 10.
%e a(4) = A046315(20)-A046315(19) = 111 - 95 = 16.
%e a(5) = A046315(55)-A046315(54) = 287 - 267 = 20.
%t f[n_] := Block[{k = n + 2}, While[ Plus @@ Last /@ FactorInteger@k != 2, k += 2]; k]; lst = {}; d = 0; a = b = 9; Do[{a, b} = {b, f[a]}; If[b - a > d, d = b - a; AppendTo[lst, a]], {n, 10^8}]; lst (* _Robert G. Wilson v_, Feb 03 2006 *)
%Y Cf. A001358, A046315, A065516, A085809, A100484, A114412, A114021.
%Y Starting at a(4)=95 the known terms of this sequence coincide with A350098.
%Y Cf. A341828, A349995, A350099.
%K nonn
%O 1,1
%A _Jonathan Vos Post_, Feb 02 2006
%E More terms from _Robert G. Wilson v_ and _T. D. Noe_, Feb 03 2006
%E a(23)-a(28) from _Donovan Johnson_, Mar 14 2010
%E a(29)-a(31) from _Donovan Johnson_, Oct 20 2012