A350419 - OEIS

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

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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

Table of n, a(n) for n=1..61.

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

EXAMPLE

Row # | m | partitions of m = s+t | semiprimes k = s*t

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1 | 4 | 4 = 2+2 --> | 2*2 = 4;

2 | 6 | 6 = 3+3 --> | 3*3 = 9;

3 | 8 | 8 = 3+5 --> | 3*5 = 15;