Concatenated Triples Of Cubes and Doubles of Squares

The number associated with my diurnal age today, 27640, has the property that it is a sum of concatentated cubes:$$ \begin{align} 27640 &= 27 \, | \,64  \,|\, 0\\ &= 3^3  \,| \, 4^3 \, |\, 0^3 \end{align}$$I then sought to discover what other numbers in the range up to 40,000 have this property. It turns out that there are 78 such numbers and they are (permalink):

108, 180, 801, 810, 1027, 1064, 1270, 1278, 1640, 1648, 1827, 1864, 2701, 2708, 2710, 2718, 2780, 2781, 6401, 6408, 6410, 6418, 6480, 6481, 8027, 8064, 8127, 8164, 8270, 8271, 8640, 8641, 10125, 10216, 10343, 10512, 10729, 11250, 11258, 12160, 12168, 12501, 12508, 12510, 12518, 12580, 12581, 12764, 13430, 13438, 15120, 15128, 16427, 17290, 17298, 18125, 18216, 18343, 18512, 18729, 21601, 21608, 21610, 21618, 21680, 21681, 27064, 27164, 27640, 27641, 27648, 27864, 34301, 34308, 34310, 34318, 34380, 34381

Let's take the last member of the above set:$$ \begin{align} 34381 &= 343 \, | \, 8 \, | \, 1\\ & =7^3 \, | \, 2^3 \, | \, 1^3 \end{align} $$It can be noted that 27640 and 27641 form a pair of consecutive numbers sharing this same property. In the range up to 40,000 this pairing occurs as shown:

• 2780 and 2781
• 6480 and 6481
• 8270 and 8271
• 8640 and 8641
• 12580 and 12581
• 21680 and 21681
• 27640 and 27641
• 34380 and 34381

If instead of cubes, we consider squares and instead of triples we consider doubles then we have the following 557 numbers in the range up to 40,000 (permalink):

10, 14, 19, 40, 41, 49, 90, 91, 94, 116, 125, 136, 149, 160, 161, 164, 164, 169, 181, 250, 251, 254, 259, 360, 361, 364, 369, 416, 425, 436, 449, 464, 481, 490, 491, 494, 499, 640, 641, 644, 649, 810, 811, 814, 819, 916, 925, 936, 949, 964, 981, 1000, 1001, 1004, 1009, 1100, 1121, 1144, 1169, 1196, 1210, 1211, 1214, 1219, 1225, 1256, 1289, 1324, 1361, 1400, 1440, 1441, 1441, 1444, 1449, 1484, 1529, 1576, 1625, 1625, 1636, 1649, 1664, 1676, 1681, 1690, 1691, 1694, 1699, 1729, 1784, 1841, 1900, 1960, 1961, 1961, 1964, 1969, 2250, 2251, 2254, 2259, 2516, 2536, 2549, 2560, 2561, 2564, 2564, 2569, 2581, 2890, 2891, 2894, 2899, 3240, 3241, 3244, 3249, 3610, 3611, 3614, 3616, 3619, 3625, 3649, 3664, 3681, 4000, 4001, 4004, 4009, 4100, 4121, 4144, 4169, 4196, 4225, 4256, 4289, 4324, 4361, 4400, 4410, 4411, 4414, 4419, 4441, 4484, 4529, 4576, 4625, 4676, 4729, 4784, 4840, 4841, 4841, 4844, 4849, 4900, 4916, 4925, 4936, 4961, 4964, 4981, 5290, 5291, 5294, 5299, 5760, 5761, 5764, 5769, 6250, 6251, 6254, 6259, 6416, 6425, 6436, 6449, 6481, 6760, 6761, 6764, 6769, 7290, 7291, 7294, 7299, 7840, 7841, 7844, 7849, 8116, 8125, 8136, 8149, 8164, 8410, 8411, 8414, 8419, 9000, 9001, 9004, 9009, 9100, 9121, 9144, 9169, 9196, 9225, 9256, 9289, 9324, 9361, 9400, 9441, 9484, 9529, 9576, 9610, 9611, 9614, 9619, 9625, 9676, 9729, 9784, 9841, 9900, 9961, 10016, 10025, 10036, 10049, 10064, 10081, 10240, 10241, 10244, 10249, 10890, 10891, 10894, 10899, 11024, 11089, 11156, 11225, 11296, 11369, 11444, 11521, 11560, 11561, 11564, 11569, 11600, 11681, 11764, 11849, 11936, 12025, 12116, 12116, 12125, 12136, 12149, 12164, 12181, 12209, 12250, 12251, 12254, 12259, 12304, 12401, 12500, 12601, 12704, 12809, 12916, 12960, 12961, 12964, 12969, 13025, 13136, 13249, 13364, 13481, 13600, 13690, 13691, 13694, 13699, 13721, 13844, 13969, 14096, 14225, 14356, 14416, 14425, 14436, 14440, 14441, 14444, 14449, 14449, 14464, 14481, 14489, 14624, 14761, 14900, 15041, 15184, 15210, 15211, 15214, 15219, 15329, 15476, 15625, 15776, 15929, 16000, 16001, 16004, 16009, 16084, 16100, 16121, 16144, 16169, 16196, 16225, 16241, 16256, 16289, 16324, 16361, 16400, 16400, 16441, 16484, 16529, 16561, 16576, 16625, 16676, 16724, 16729, 16784, 16810, 16811, 16814, 16819, 16841, 16889, 16900, 16916, 16925, 16936, 16949, 16961, 16964, 16981, 17056, 17225, 17396, 17569, 17640, 17641, 17644, 17649, 17744, 17921, 18100, 18281, 18464, 18490, 18491, 18494, 18499, 18649, 18836, 19025, 19216, 19360, 19361, 19364, 19369, 19409, 19604, 19616, 19625, 19636, 19649, 19664, 19681, 19801, 20250, 20251, 20254, 20259, 21160, 21161, 21164, 21169, 22090, 22091, 22094, 22099, 22516, 22525, 22536, 22549, 22564, 22581, 23040, 23041, 23044, 23049, 24010, 24011, 24014, 24019, 25000, 25001, 25004, 25009, 25100, 25121, 25144, 25169, 25196, 25225, 25256, 25289, 25324, 25361, 25400, 25441, 25484, 25529, 25576, 25616, 25625, 25625, 25636, 25649, 25664, 25676, 25681, 25729, 25784, 25841, 25900, 25961, 26010, 26011, 26014, 26019, 27040, 27041, 27044, 27049, 28090, 28091, 28094, 28099, 28916, 28925, 28936, 28949, 28964, 28981, 29160, 29161, 29164, 29169, 30250, 30251, 30254, 30259, 31360, 31361, 31364, 31369, 32416, 32425, 32436, 32449, 32464, 32481, 32490, 32491, 32494, 32499, 33640, 33641, 33644, 33649, 34810, 34811, 34814, 34819, 36000, 36001, 36004, 36009, 36100, 36116, 36121, 36125, 36136, 36144, 36149, 36164, 36169, 36181, 36196, 36225, 36256, 36289, 36324, 36361, 36400, 36441, 36484, 36529, 36576, 36625, 36676, 36729, 36784, 36841, 36900, 36961, 37210, 37211, 37214, 37219, 38440, 38441, 38444, 38449, 39690, 39691, 39694, 39699

Let's take the last number in the above set of numbers:$$ \begin{align} 39699 &= 3969 \, | \, 9 \\ &= 63^2 \, | \, 3^2 \end{align}$$One more example is:$$ \begin{align} 36000 &= 3600 \, | \, 0 \\ &= 60^2 \, | \, 0^2 \end{align} $$There are obviously many other variation on these themes but that will suffice for now.