A241535 Smallest semiprime q such that 2*prime(n) - q is semiprime, or a(n)=0 if there is no such q.

0, 0, 4, 4, 0, 4, 9, 4, 21, 9, 4, 9, 25, 4, 9, 15, 25, 4, 15, 9, 4, 15, 21, 9, 9, 15, 4, 9, 4, 9, 33, 9, 9, 4, 9, 4, 9, 21, 15, 25, 35, 4, 21, 4, 33, 4, 9, 9, 9, 4, 15, 9, 4, 9, 9, 9, 9, 4, 9, 9, 4, 21, 25, 25, 4, 51, 33, 25, 9, 4, 9, 15, 21, 9, 9, 21, 15, 9, 9, 15, 21, 4, 21, 4

Offset: 1

Author

Vladimir Shevelev, Apr 25 2014

Keywords

• nonn

Comments

Conjecture: every even semiprime more than 22 is a sum of two semiprimes.

All terms are either 4 or odd.

First occurrence of k-th odd semiprime (A046315): 7, 16, 9, 13, 31, 41, 104, 134, 66, 412, 2769, 447, 1282, 3868, 7003, 3601, 48649, 11016, 5379, 41644, 34575, 83474, 120165, 135566, 21335, 394140, 14899, 876518, 434986, 173914, 691409, 1854580, 3741206, 714807, 1001321, 6427837, 4267513, 14809496, 7795998, 26617567, 2001937, 13958857, 9217135, 18815676, ..., . - Robert G. Wilson v, Apr 26 2014

If a(n) = 4, then 2*prime(n)-4=2*(prime(n)-2) is a semiprime, thus prime(n)-2 is a prime, so prime(n) belongs to A006512 (greater of twin primes). - Michel Marcus, Mar 26 2015

Examples

			Let n=16, then 2*prime(16) = 2*53 = 106. We have 106-4=102, 106-6=100, 106-9=97, 106-10=96, 106-14=92, 106-15=91 and only the last number is semiprime. So a(16)=15.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001358, A241531, A241533.

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[{p2 = 2 Prime[n], sp = 4}, While[ PrimeOmega[p2 - sp] != 2, sp = NextSemiPrime[sp]]; If[ sp != p2, sp, 0]]; Array[f, 75] (* Robert G. Wilson v, Apr 25 2014 *)

Extensions

More terms from Robert G. Wilson v, Apr 25 2014