1,1

5 of the first 6 values of record gaps in even semiprimes are also record merits = [A100484(k+1)-A100484(k)]/log(A100484(k)], namely: (6 - 4) / log(4) = 3.32192809; (10 - 6) / log(6) = 5.14038884; (22 - 14) / log(14) = 6.98002296; (58 - 46) / log(46) = 7.21692586; (254 - 226) / log(226) = 11.8940995. It is easy to prove that there are gaps of arbitrary length in even semiprimes (A100484), as 2*(n!+2), 2*(n!+3), 2*(n!+4), ..., 2*(n!+n) gives (n-1) consecutive even nonsemiprimes. Can we prove that there are gaps of arbitrary length in odd semiprimes (A046315), and in semiprimes (A001358)?

a(n) = A100484(k) such that A100484(k+1)-A100484(k) is a record.

gap[a(1)] = A100484(2)-A100484(1) = 6 - 4 = 2.

gap[a(2)] = A100484(3)-A100484(2) = 10 - 6 = 4.

gap[a(3)] = A100484(5)-A100484(4) = 22 - 14 = 8.

gap[a(4)] = A100484(10)-A100484(9) = 58 - 46 = 12.

gap[a(5)] = A100484(25)-A100484(24) = 194 - 178 = 16.

gap[a(6)] = A100484(31)-A100484(30) = 254 - 226 = 28.

f[n_] := Block[{k = n + 2}, While[ Plus @@ Last /@ FactorInteger@k != 2, k += 2]; k]; lst = {}; d = 0; a = b = 4; 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.

Sequence in context: A108516 A077068 A096003 this_sequence A135093 A141667 A048753

Adjacent sequences: A114055 A114056 A114057 this_sequence A114059 A114060 A114061

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 02 2006

a(7)-a(25) from Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 03 2006