Palindromic Day 27272

It's a palindromic day again, which happens every one hundred days during my current millennium (27000 to 27999). The number associated with my diurnal age today (27272) has a connection to triangular numbers, a topic that I wrote about in two recent posts titled Happy Triangular Numbers on the 22nd November 2023 and Four Fun Facts About Triangular Numbers on the 25th November 2023.

The connection of 27272 to triangular numbers arises via OEIS A340953:

A340953

Number of ways to write \(n\) as an ordered sum of eight nonzero triangular numbers.

So when \(n=55\) it turns out that it can be written as an ordered sum of eight nonzero triangular numbers in 27272 different ways. The triangular numbers less or equal to 55 are as follows:$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55$$Using these numbers, and only these number, one possible sum would be:$$3 + 3 + 6 + 6 + 6 + 6 +10 + 15 = 55$$What makes the number of possibilities so high is that the order of the terms in the sum is being taken into account.

The initial members of the sequence are (permalink):

1, 0, 8, 0, 28, 8, 56, 56, 70, 176, 84, 336, 196, 448, 492, 504, 953, 616, 1456, 960, 1814, 1792, 1904, 3032, 2100, 4144, 3052, 4768, 4670, 5264, 6720, 5936, 8876, 7112, 10620, 9648, 11718, 12720, 13216, 15960, 15261, 19608, 17164, 23296, 21226, 25424, 26796, 27272, 32844, 30480, 38640, 34160, 43512

Take the case of \(n=10\). There are eight possible ordered sums and they are:$$1+1+1+1+1+1+1+3 =55\\1+1+1+1+1+1+3+1 =55\\1+1+1+1+1+3+1+1 =55\\1+1+1+1+3+1+1+1 =55\\1+1+1+3+1+1+1+1 =55\\1+1+3+1+1+1+1+1 =55\\1+3+1+1+1+1+1+1 =55\\3+1+1+1+1+1+1+3 =55 $$Another property of 27272 that is not immediately obvious is that it stands in the middle of a run of so-called iban numbers. These are numbers that do not contain the letter "i" when written using the letters of the alphabet. The run is:$$27270, 27271, 27272, 27273, 27274$$I've written about these in my post titled Iban Numbers on July 31st 2023. In the case of 27272, it is written as:

Two Thousand Two Hundred Seventy Two

What other interesting properties does 27272 have? Well, for one, it belongs to a sequence of composite numbers whose arithmetic derivatives have no digits in common with the originating number. $$ \begin{align} 27272 &= 2 \times 2 \times 2 \times 7 \times 487 \\ 27272' &= 44860 \end{align}$$Here is a permalink to the calculation. 27272 is also what is called a nude number because it divisible by every one of its digits. However, if we look more closely, we see that the number is also divisible by every one of the digits in its prime factors (2, 4, 7 and 8).$$27272=2^3 * 7 * 487$$There are only 337 such numbers in the range up to 40,000 (none seem to end in 3, 7 or 9 except for the initial single digit 9) and they are:

1, 4, 6, 8, 9, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 112, 126, 128, 132, 135, 144, 162, 168, 175, 216, 224, 264, 288, 312, 315, 324, 336, 366, 384, 396, 432, 448, 624, 648, 672, 735, 777, 784, 864, 936, 1116, 1155, 1176, 1197, 1248, 1266, 1296, 1344, 1368, 1395, 1448, 1464, 1488, 1575, 1715, 1764, 1848, 1944, 2112, 2184, 2196, 2232, 2248, 2688, 2744, 2772, 2916, 3132, 3144, 3168, 3276, 3312, 3432, 3444, 3612, 3864, 3888, 4116, 4128, 4212, 4224, 4344, 4368, 4392, 4416, 4464, 4644, 4968, 5115, 5355, 5775, 6132, 6144, 6192, 6216, 6264, 6288, 6312, 6336, 6624, 6696, 6762, 6864, 6888, 6912, 7112, 7119, 7224, 7266, 7371, 7644, 7728, 8112, 8136, 8184, 8232, 8424, 8448, 8688, 8736, 8832, 8928, 9126, 9216, 9288, 9324, 9396, 9432, 9936, 11112, 11115, 11184, 11196, 11232, 11316, 11424, 11616, 11664, 11848, 11916, 12144, 12168, 12222, 12264, 12288, 12312, 12366, 12384, 12432, 12624, 12636, 12666, 12712, 12816, 12996, 13122, 13248, 13326, 13377, 13392, 13416, 13488, 13662, 13755, 13776, 13797, 13824, 13896, 13932, 13995, 14112, 14224, 14328, 14364, 14448, 14488, 14616, 14784, 16128, 16164, 16224, 16236, 16332, 16368, 16416, 16464, 16488, 16632, 16848, 17136, 17199, 17248, 17262, 17472, 17724, 17955, 18144, 18216, 18248, 18384, 18424, 18432, 18648, 18816, 18864, 18936, 19224, 19368, 19719, 19971, 21144, 21168, 21222, 21264, 21336, 21384, 21492, 21648, 21672, 21888, 21924, 22122, 22128, 22176, 22224, 22326, 22368, 22392, 22464, 22632, 22764, 22848, 22932, 22968, 23136, 23166, 23184, 23232, 23322, 23328, 23424, 23436, 23616, 23688, 23832, 24192, 24276, 24288, 24336, 24444, 24624, 24696, 24864, 26124, 26136, 26244, 26364, 26496, 26832, 27216, 27272, 27384, 27636, 27762, 27888, 27972, 28224, 28296, 28344, 28448, 28728, 29232, 29412, 31122, 31248, 31266, 31311, 31332, 31416, 31464, 31488, 31644, 31896, 32112, 32184, 32292, 32328, 32364, 32448, 32664, 32832, 33144, 33192, 33222, 33264, 33336, 33444, 33696, 33726, 33768, 34224, 34272, 34416, 34776, 34848, 34992, 35595, 36126, 36288, 36372, 36432, 36612, 36666, 36792, 36864, 36936, 37128, 37212, 37296, 37317, 37464, 37632, 38136, 38448, 38688, 39312, 39366, 39375, 39816