A350419 - OEIS

%I #33 Feb 16 2025 08:34:02

%S 4,9,15,9,21,25,35,33,49,15,39,55,65,77,51,91,21,57,85,121,95,119,143,

%T 25,69,133,169,115,187,161,209,221,87,247,33,93,145,253,289,35,155,

%U 203,299,323,217,361,39,111,319,391,185,341,377,437,123,259,403,129,205,493,529

%N 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.

%C 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.

%C 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).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

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

%e -----------------------------------------------------------------------

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

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

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

%e    4     | 10  |  10 = 1+9 = 3+7 = 5+5 -->  |   1*9 = 9, 3*7 = 21, 5*5 = 25;

%e    5     | 12  |  12 = 5+7 -->              |   5*7 = 35;

%e    6     | 14  |  14 = 3+11 = 7+7 -->       |   3*11 = 33, 7*7 = 49;

%e ...

%p T:= n-> select(x-> numtheory[bigomega](x)=2, [seq(s*(2*n+2-s), s=1..n+1)])[]:

%p seq(T(n), n=1..22);  # _Alois P. Heinz_, Dec 31 2021

%Y Cf. A001358, A105020.

%K nonn,tabf,changed

%O 1,1

%A _Wesley Ivan Hurt_, Dec 29 2021