1,1

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)?

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 (noe(AT)sspectra.com), Feb 03 2006

{a(n)} = {A046315(k) such that A046315(k+1)-A046315(k) is a record}.

a(1) = A046315(2)-A046315(1) = 15 - 9 = 6.

a(2) = A046315(5)-A046315(4) = 33 - 25 = 8.

a(3) = A046315(8)-A046315(7) = 49 - 39 = 10.

a(4) = A046315(20)-A046315(19) = 111 - 95 = 16.

a(5) = A046315(55)-A046315(54) = 287 - 267 = 20.

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 *)

Cf. A001358, A046315, A065516, A085809, A100484, A114412, A114021.

Adjacent sequences: A114054 A114055 A114056 this_sequence A114058 A114059 A114060

Sequence in context: A137190 A044070 A044451 this_sequence A031036 A051132 A075026

easy,nonn

Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 02 2006

More terms from Robert G. Wilson v (rgwv(at)rgwv.com) and T. D. Noe (noe(AT)sspectra.com), Feb 03 2006