A339709 a(n) is the least even number that has exactly n decompositions as the sum of an odd prime and a semiprime, or 0 if there is no such number.

2, 12, 20, 26, 28, 44, 38, 52, 76, 80, 62, 92

Offset: 0

Author

J. M. Bergot and Robert Israel, Dec 13 2020

Keywords

• nonn
• more

Comments

a(n) is the least even number 2*k such that A339708(k)=n.

Conjecture: a(12) = 0, but a(n) > 0 for all other n.

Since this is only a conjecture, the Data stops at a(11)=92.

For n>=13, the sequence continues with 98, 136, 128, 122, 158, 166, 152, 206, 188, 222, 242, 232, 248, 266, 272, 296, 320, 308, 352, 382, 412, 326, 418, 402, 440, 454, 398, 492, 458, 488, 500, 554, 542, 518, 578, 618, 572, 626, 642, 678, 632, ...

Examples

			a(4) = 28 because we can write 28 = 3+25 = 7+21 = 13+15 = 19+9 where 3, 7, 13 and 19 are odd primes and 25, 21, 15, and 9 are semiprimes, and 28 is the least even number with exactly 4 decompositions of this type.
		

Links

• Robert Israel, Table of n, a(n) for n = 0 .. 800 (with conjectured a(12)=0).

Crossrefs

Cf. A339708.

Programs

Maple

N:= 10000:
P:= select(isprime,[seq(i,i=3..N,2)]):S:= sort(select(`<`,[seq(seq(P[i]*P[j],i=1..j),j=1..nops(P))],N)):V:= Vector(N):
for p in P do
  for s in S while p+s<=N do V[p+s]:= V[p+s]+1 od
od:R:= Array(0..max(V)):for i from 2 to N by 2 do
  v:= V[i];
  if R[v] = 0 then R[v]:= i fi
od:convert(R[1..500],list);