A241658 - OEIS

%I #16 Mar 28 2015 16:17:27

%S 0,0,0,0,0,0,0,4,0,4,0,6,4,4,6,6,0,4,4,6,6,0,9,9,4,4,6,6,4,4,6,6,0,9,

%T 9,10,4,4,4,6,6,4,4,6,6,21,9,9,10,4,25,6,4,15,4,10,6,9,4,9,4,4,6,6,10,

%U 4,9,6,4,15,6,10,4,9,6,14,15,4,10,6,4,25,6,10,34,4,10,6,4,4

%N Smallest semiprime, sp, such that n - sp is a semiprime, or a(n)=0 if there is no such sp.

%C Conjecture: every number greater than 33 is a sum of two semiprimes. Only 1, 2, 3, 4, 5, 6, 7, 9, 11, 17, 22 & 33 cannot be so represented.

%C If n is prime, then a(2n) must be either 4 or an odd semiprime. See A241535.

%C First occurrence of the k-th semiprime (A001358): 8, 12, 23, 36, 76, 54, 46, 113, 51, 185, 254, 85, 294, 1881, 378, 1035, 1514, 634, 1509, 3550, 1621, 2713, 4050, 14788, 1485, 26839, 1497, 22694, 11965, 15334, 15810, 30894, 2721, 16849, ..., .

%e a(23) = 9 because 23 = 9 + 14, 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[{sp = 4}, While[ PrimeOmega[n - sp] != 2, sp = NextSemiPrime[sp]]; If[n > sp, sp, 0]]; Array[ f, 90]

%o (PARI) a(n) = {for (k=4, n-4, if ((bigomega(k) ==2) && (bigomega(n-k) == 2), return (k));); return (0);} \\ _Michel Marcus_, Jun 12 2014

%Y Cf. A001358, A241535, A241656.

%K nonn

%O 1,8

%A _Vladimir Shevelev_ and _Robert G. Wilson v_, Apr 26 2014