OFFSET
1,1
COMMENTS
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)?
For every n, a(n) = 2*A002386(n). [John W. Nicholson, Jul 26 2012]
LINKS
John W. Nicholson, Table of n, a(n) for n = 1..75
EXAMPLE
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 02 2006
EXTENSIONS
a(7)-a(25) from Robert G. Wilson v, Feb 03 2006
a(26)-a(31) from Donovan Johnson, Mar 14 2010
STATUS
approved