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 A114057 Start of record gap in odd semiprimes A046315. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 A348749 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

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Last modified March 22 18:49 EDT 2023. Contains 361433 sequences. (Running on oeis4.)