OFFSET
1,1
COMMENTS
The sequence consists of the set {4} UNION {odd semiprimes}. Every odd semiprime in the sequence appears exactly twice since for each partition of m = s + t where s, t are prime, there exists another partition of the form 1 + s*t and vice versa.
If the Goldbach conjecture is true, each row of the table in the example will have at least one Goldbach partition, m = s + t, where s and t are prime. For each odd semiprime that makes its first appearance in the sequence, and thus in some row u = m/2-1 of the table, that semiprime will occur again exactly once in row v = (s*t-1)/2 as the partition 1 + s*t. Likewise, each odd semiprime that makes its second appearance in the sequence will be a partition of some m of the form s + t = 1 + pq in some row v where p and q are (odd) primes. Its first occurrence will appear earlier in row u = (p+q)/2-1 of the table (see example).
LINKS
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
EXAMPLE
Row # | m | partitions of m = s+t | semiprimes k = s*t
-----------------------------------------------------------------------
1 | 4 | 4 = 2+2 --> | 2*2 = 4;
2 | 6 | 6 = 3+3 --> | 3*3 = 9;
3 | 8 | 8 = 3+5 --> | 3*5 = 15;
4 | 10 | 10 = 1+9 = 3+7 = 5+5 --> | 1*9 = 9, 3*7 = 21, 5*5 = 25;
5 | 12 | 12 = 5+7 --> | 5*7 = 35;
6 | 14 | 14 = 3+11 = 7+7 --> | 3*11 = 33, 7*7 = 49;
...
MAPLE
T:= n-> select(x-> numtheory[bigomega](x)=2, [seq(s*(2*n+2-s), s=1..n+1)])[]:
seq(T(n), n=1..22); # Alois P. Heinz, Dec 31 2021
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Wesley Ivan Hurt, Dec 29 2021
STATUS
approved