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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 (list; graph; refs; listen; history; text; internal format)
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

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

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Last modified March 26 01:08 EDT 2019. Contains 321479 sequences. (Running on oeis4.)