

A114058


Start of record gap in even semiprimes (A100484).


1



4, 6, 14, 46, 178, 226, 1046, 1774, 2258, 2654, 19102, 31366, 39218, 62794, 311842, 721306, 740522, 984226, 2699066, 2714402, 4021466, 9304706, 34103414, 41662646, 94653386, 244329494, 379391318, 383825566, 774192266
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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 (n1) 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


FORMULA

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


EXAMPLE

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.


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

Cf. A001358, A046315, A065516, A085809, A100484, A114412, A114021. Maximal gap small prime A002386.
Sequence in context: A077068 A277059 A096003 * A214901 A284123 A135093
Adjacent sequences: A114055 A114056 A114057 * A114059 A114060 A114061


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



