A350419 Irregular table read by rows, where row k lists the semiprimes, s*t (s<=t) in increasing order, where s and t are the smaller and larger parts of the partitions of m = 2k+2 into two parts.

4, 9, 15, 9, 21, 25, 35, 33, 49, 15, 39, 55, 65, 77, 51, 91, 21, 57, 85, 121, 95, 119, 143, 25, 69, 133, 169, 115, 187, 161, 209, 221, 87, 247, 33, 93, 145, 253, 289, 35, 155, 203, 299, 323, 217, 361, 39, 111, 319, 391, 185, 341, 377, 437, 123, 259, 403, 129, 205, 493, 529

Offset: 1

Author

Wesley Ivan Hurt, Dec 29 2021

Keywords

• nonn
• tabf

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).

Examples

			  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;
...
		

Links

• Eric Weisstein's World of Mathematics, Goldbach Partition
• Wikipedia, Goldbach's conjecture
• Index entries for sequences related to Goldbach conjecture
• Index entries for sequences related to partitions

Crossrefs

Cf. A001358, A105020.

Programs

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