|
%I #17 May 03 2014 01:09:49
%S 0,0,0,4,4,6,4,6,4,6,0,9,4,6,4,6,9,10,4,6,4,6,21,9,4,6,15,10,9,9,4,6,
%T 4,6,15,10,9,14,4,6,25,10,4,6,4,6,9,9,4,6,9,9,15,14,4,6,21,10,25,9,4,
%U 6,4,6,9,9,15,14,4,6,9,10,4,6,4,6,9,10,15,14,4,6,21,9,4,6,15,10,9,14
%N Smallest semiprime, sp, such that 2n - sp is a semiprime, or a(n)=0 if there is no such sp.
%C Conjecture: every even number greater than 22 is a sum of two semiprimes. Only 2, 4, 6 & 22 cannot be so represented.
%C If n is prime, then a(n) must be either 4 or an odd semiprime. See A241535.
%C First occurrence of the k-th semiprime (A001358): 4, 6, 12, 18, 38, 27, 23, 124, 41, 326, 127, 1344, 147, 1278, 189, 3294, 757, 317, 1362, 1775, 3300, 2504, 2025, 7394, 84848, 13899, 56584, 11347, 156396, 7667, 7905, 15447, 404898, 20937, ..., .
%e a(12) = 9 because 2*12 = 24 = 9 + 15, two semiprimes.
%t NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[ sgn < 0, sp--, sp++]]; If[ sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Block[{en = 2 n, sp = 4}, While[ PrimeOmega[en - sp] != 2, sp = NextSemiPrime[sp]]; If[en > sp, sp, 0]]; Array[ f, 90]
%Y Cf. A001358, A241535, A241658.
%K nonn
%O 1,4
%A _Vladimir Shevelev_ and _Robert G. Wilson v_, Apr 26 2014
|
