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

1,1

Comments

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, Feb 03 2006

Links

Table of n, a(n) for n=1..31.

Formula

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

Example

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.

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 *)

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.

Sequence in context: A147403 A241764 A044451 * A227518 A031036 A371086

Adjacent sequences: A114054 A114055 A114056 * A114058 A114059 A114060

Keyword

nonn

Author

Jonathan Vos Post, Feb 02 2006

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

Status

approved