A241656 Smallest semiprime, sp, such that 2n - sp is a semiprime, or a(n)=0 if there is no such sp.

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, 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, 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

Offset: 1

Author

Vladimir Shevelev and Robert G. Wilson v, Apr 26 2014

Keywords

• nonn

Comments

Conjecture: every even number greater than 22 is a sum of two semiprimes. Only 2, 4, 6 & 22 cannot be so represented.

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

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, ..., .

Examples

			a(12) = 9 because 2*12 = 24 = 9 + 15, two semiprimes.
		

Crossrefs

Cf. A001358, A127018, A241535, A241658.

Programs

Mathematica

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]