The Search for Patterns

Today I turned 26338 days old and a search of the OEIS turned up only two entries, neither of them meaningful to me. Further searching didn't turn up anything more. The factorisation of the number is $2 \times 13 \times 1013$ and of course it struck me that $2 \times 13 = 26$ which represents the first two digits of the number. It was some time later that I considered the last three digits and realised that $338 = 2 \times 13^2$.

Suddenly there was a pattern and where there's a pattern, there's a sequence. The pattern is to concatenate $2n$ and $2n^2$. After that, it didn't take too long to create the sequence. See Figure 1.

Figure 1: permalink

Thus the sequence of terms up to 26338 is: 

22, 48, 618, 832, 1050, 1272, 1498, 16128, 18162, 20200, 22242, 24288, 26338

There is a similar sequence in the OEIS viz. A053061

A053061

a($n$) is the decimal concatenation of $n$ and $n^2$.

The terms in this sequence run:

1, 24, 39, 416, 525, 636, 749, 864, 981, 10100, 11121, 12144, 13169, 14196, 15225, 16256, 17289, 18324, 19361, 20400, 21441, 22484, 23529, 24576, 25625, 26676, 27729, 28784, 29841, 30900, 31961, 321024, 331089, 341156, 351225, 361296, 371369, 381444, 391521 

So my new sequence would appear as:

A******

a($n$) is the decimal concatenation of $2n$ and $2n^2$.

I could submit it for consideration by the OEIS arbiters but I probably won't as I find dealing with them rather tedious. However, I might change my mind. The sequence could be generalised of course to something like:

A******

a($n$) is the decimal concatenation of $kn$ and $kn^2$ where $k = 2$

I was happy to discover the concatenation of $2n$ and $2n^2$ in regard to 26338 as it shows that there are often patterns lurking in numbers that at first sight are not apparent. The list of terms, up to $n=100$ is:

22, 48, 618, 832, 1050, 1272, 1498, 16128, 18162, 20200, 22242, 24288, 26338, 28392, 30450, 32512, 34578, 36648, 38722, 40800, 42882, 44968, 461058, 481152, 501250, 521352, 541458, 561568, 581682, 601800, 621922, 642048, 662178, 682312, 702450, 722592, 742738, 762888, 783042, 803200, 823362, 843528, 863698, 883872, 904050, 924232, 944418, 964608, 984802, 1005000, 1025202, 1045408, 1065618, 1085832, 1106050, 1126272, 1146498, 1166728, 1186962, 1207200, 1227442, 1247688, 1267938, 1288192, 1308450, 1328712, 1348978, 1369248, 1389522, 1409800, 14210082, 14410368, 14610658, 14810952, 15011250, 15211552, 15411858, 15612168, 15812482, 16012800, 16213122, 16413448, 16613778, 16814112, 17014450, 17214792, 17415138, 17615488, 17815842, 18016200, 18216562, 18416928, 18617298, 18817672, 19018050, 19218432, 19418818, 19619208, 19819602, 20020000

Of course, there can be no primes in this sequence because $2 \times n$ will always be a factor and this is most easily seen when $n$ is prime e.g. for $n=83$, the sequence term is 16613778 which factorises to $2 \times 83 \times 3 \times 73 \times 457$.