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 \(2 \times 83 \times 3 \times 73 \times 457\).