A114057 Start of record gap in odd semiprimes A046315.

9, 25, 39, 95, 267, 2369, 6559, 8817, 13705, 15261, 21583, 35981, 66921, 113009, 340891, 783757, 872219, 3058853, 3586843, 5835191, 12345473, 108994623, 248706917, 268749691, 679956119, 709239621, 3648864859, 3790337723, 4171420481, 33955869693, 34279038379

Offset: 1

Author

Jonathan Vos Post, Feb 02 2006

Keywords

• nonn

Comments

3 of the first 5 values of record gaps in odd semiprimes are also record merits = (A046315(k+1)-A046315(k))/log_10(A046315(k)), namely: (15 - 9) / log_10(9) = 6.28770982; (111 - 95) / log_10(95) = 8.09010923; (287 - 267) / log_10(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, Feb 03 2006

Examples

			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.
		

Crossrefs

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

Starting at a(4)=95 the known terms of this sequence coincide with A350098.

Cf. A341828, A349995, A350099.

Programs

Mathematica

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 *)

Formula

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

Extensions

More terms from Robert G. Wilson v and T. D. Noe, Feb 03 2006

a(23)-a(28) from Donovan Johnson, Mar 14 2010

a(29)-a(31) from Donovan Johnson, Oct 20 2012