Concatenations of Squares and Cubes

I was surprised that the number associated with my diurnal age today, 27125, didn't show up in the OEIS as a concatenation of two cubes, namely \(3^3\) and \(5^3\). This prompted me to list all numbers of the form \(n^3 \, | \, m^3\) where \(n\) and \(m\) are integers (not necessarly distinct). Here is a permalink to the SageMath code that generates the list up to 40,000 and here is the list:

11, 18, 81, 88, 127, 164, 271, 278, 641, 648, 827, 864, 1125, 1216, 1251, 1258, 1343, 1512, 1729, 2161, 2168, 2727, 2764, 3431, 3438, 5121, 5128, 6427, 6464, 7291, 7298, 8125, 8216, 8343, 8512, 8729, 10001, 10008, 11000, 11331, 11728, 12197, 12527, 12564, 12744, 13311, 13318, 13375, 14096, 14913, 15832, 16859, 17281, 17288, 18000, 19261, 21627, 21664, 21971, 21978, 27125, 27216, 27343, 27441, 27448, 27512, 27729, 33751, 33758, 34327, 34364

Some are more difficult to spot than others. What about concatenations of square numbers? Here is a permalink to the SageMath code that generates a list of numbers up to 40,000 and here is the list of numbers of the form  \(n^2 \, | \, m^2\) where \(n\) and \(m\) are integers (not necessarly distinct):

11, 14, 19, 41, 44, 49, 91, 94, 99, 116, 125, 136, 149, 161, 164, 169, 181, 251, 254, 259, 361, 364, 369, 416, 425, 436, 449, 464, 481, 491, 494, 499, 641, 644, 649, 811, 814, 819, 916, 925, 936, 949, 964, 981, 1001, 1004, 1009, 1100, 1121, 1144, 1169, 1196, 1211, 1214, 1219, 1225, 1256, 1289, 1324, 1361, 1400, 1441, 1444, 1449, 1484, 1529, 1576, 1616, 1625, 1636, 1649, 1664, 1676, 1681, 1691, 1694, 1699, 1729, 1784, 1841, 1900, 1961, 1964, 1969, 2251, 2254, 2259, 2516, 2525, 2536, 2549, 2561, 2564, 2569, 2581, 2891, 2894, 2899, 3241, 3244, 3249, 3611, 3614, 3616, 3619, 3625, 3636, 3649, 3664, 3681, 4001, 4004, 4009, 4100, 4121, 4144, 4169, 4196, 4225, 4256, 4289, 4324, 4361, 4400, 4411, 4414, 4419, 4441, 4484, 4529, 4576, 4625, 4676, 4729, 4784, 4841, 4844, 4849, 4900, 4916, 4925, 4936, 4949, 4961, 4964, 4981, 5291, 5294, 5299, 5761, 5764, 5769, 6251, 6254, 6259, 6416, 6425, 6436, 6449, 6464, 6481, 6761, 6764, 6769, 7291, 7294, 7299, 7841, 7844, 7849, 8116, 8125, 8136, 8149, 8164, 8181, 8411, 8414, 8419, 9001, 9004, 9009, 9100, 9121, 9144, 9169, 9196, 9225, 9256, 9289, 9324, 9361, 9400, 9441, 9484, 9529, 9576, 9611, 9614, 9619, 9625, 9676, 9729, 9784, 9841, 9900, 9961, 10016, 10025, 10036, 10049, 10064, 10081, 10241, 10244, 10249, 10891, 10894, 10899, 11024, 11089, 11156, 11225, 11296, 11369, 11444, 11521, 11561, 11564, 11569, 11600, 11681, 11764, 11849, 11936, 12025, 12116, 12125, 12136, 12149, 12164, 12181, 12209, 12251, 12254, 12259, 12304, 12401, 12500, 12601, 12704, 12809, 12916, 12961, 12964, 12969, 13025, 13136, 13249, 13364, 13481, 13600, 13691, 13694, 13699, 13721, 13844, 13969, 14096, 14225, 14356, 14416, 14425, 14436, 14441, 14444, 14449, 14464, 14481, 14489, 14624, 14761, 14900, 15041, 15184, 15211, 15214, 15219, 15329, 15476, 15625, 15776, 15929, 16001, 16004, 16009, 16084, 16100, 16121, 16144, 16169, 16196, 16225, 16241, 16256, 16289, 16324, 16361, 16400, 16441, 16484, 16529, 16561, 16576, 16625, 16676, 16724, 16729, 16784, 16811, 16814, 16819, 16841, 16889, 16900, 16916, 16925, 16936, 16949, 16961, 16964, 16981, 17056, 17225, 17396, 17569, 17641, 17644, 17649, 17744, 17921, 18100, 18281, 18464, 18491, 18494, 18499, 18649, 18836, 19025, 19216, 19361, 19364, 19369, 19409, 19604, 19616, 19625, 19636, 19649, 19664, 19681, 19801, 20251, 20254, 20259, 21161, 21164, 21169, 22091, 22094, 22099, 22516, 22525, 22536, 22549, 22564, 22581, 23041, 23044, 23049, 24011, 24014, 24019, 25001, 25004, 25009, 25100, 25121, 25144, 25169, 25196, 25225, 25256, 25289, 25324, 25361, 25400, 25441, 25484, 25529, 25576, 25616, 25625, 25636, 25649, 25664, 25676, 25681, 25729, 25784, 25841, 25900, 25961, 26011, 26014, 26019, 27041, 27044, 27049, 28091, 28094, 28099, 28916, 28925, 28936, 28949, 28964, 28981, 29161, 29164, 29169, 30251, 30254, 30259, 31361, 31364, 31369, 32416, 32425, 32436, 32449, 32464, 32481, 32491, 32494, 32499, 33641, 33644, 33649, 34811, 34814, 34819, 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, 37211, 37214, 37219, 38441, 38444, 38449, 39691, 39694, 39699

We can thin the above list of numbers by requiring that the number formed by the concatenation be a square number (permalink):

49, 169, 361, 1225, 1444, 1681, 3249, 4225, 4900, 15625, 16900, 36100

Here we see that \(36100 = 6^2 \, | \, 10^2 = 190^2 \).

While we're at it, let's consider concatenations of fourth powers. Here is a list (permalink) of numbers of the form \(n^4 \, | \, m^4\) where \(n\) and \(m\) are integers (not necessarly distinct):

11, 116, 161, 181, 811, 1256, 1616, 1625, 1681, 2561, 6251, 8116, 8181, 11296, 12401, 12961, 14096, 16256, 16561, 16625, 24011, 25616, 25681

We don't have to limit ourselves to concatenations of pairs of powers. We can concatenate three powers as easily as two. Let's consider numbers that are a concatenation of three square numbers (permalink):

111, 114, 119, 141, 144, 149, 191, 194, 199, 411, 414, 419, 441, 444, 449, 491, 494, 499, 911, 914, 919, 941, 944, 949, 991, 994, 999, 1116, 1125, 1136, 1149, 1161, 1164, 1169, 1181, 1251, 1254, 1259, 1361, 1364, 1369, 1416, 1425, 1436, 1449, 1464, 1481, 1491, 1494, 1499, 1611, 1614, 1619, 1641, 1644, 1649, 1691, 1694, 1699, 1811, 1814, 1819, 1916, 1925, 1936, 1949, 1964, 1981, 2511, 2514, 2519, 2541, 2544, 2549, 2591, 2594, 2599, 3611, 3614, 3619, 3641, 3644, 3649, 3691, 3694, 3699, 4116, 4125, 4136, 4149, 4161, 4164, 4169, 4181, 4251, 4254, 4259, 4361, 4364, 4369, 4416, 4425, 4436, 4449, 4464, 4481, 4491, 4494, 4499, 4641, 4644, 4649, 4811, 4814, 4819, 4911, 4914, 4916, 4919, 4925, 4936, 4941, 4944, 4949, 4964, 4981, 4991, 4994, 4999, 6411, 6414, 6419, 6441, 6444, 6449, 6491, 6494, 6499, 8111, 8114, 8119, 8141, 8144, 8149, 8191, 8194, 8199, 9116, 9125, 9136, 9149, 9161, 9164, 9169, 9181, 9251, 9254, 9259, 9361, 9364, 9369, 9416, 9425, 9436, 9449, 9464, 9481, 9491, 9494, 9499, 9641, 9644, 9649, 9811, 9814, 9819, 9916, 9925, 9936, 9949, 9964, 9981, 10011, 10014, 10019, 10041, 10044, 10049, 10091, 10094, 10099, 11001, 11004, 11009, 11100, 11121, 11144, 11169, 11196, 11211, 11214, 11219, 11225, 11256, 11289, 11324, 11361, 11400, 11441, 11444, 11449, 11484, 11529, 11576, 11616, 11625, 11636, 11649, 11664, 11676, 11681, 11691, 11694, 11699, 11729, 11784, 11841, 11900, 11961, 11964, 11969, 12111, 12114, 12119, 12141, 12144, 12149, 12191, 12194, 12199, 12251, 12254, 12259, 12516, 12525, 12536, 12549, 12561, 12564, 12569, 12581, 12891, 12894, 12899, 13241, 13244, 13249, 13611, 13614, 13616, 13619, 13625, 13636, 13649, 13664, 13681, 14001, 14004, 14009, 14100, 14121, 14144, 14169, 14196, 14225, 14256, 14289, 14324, 14361, 14400, 14411, 14414, 14419, 14441, 14444, 14449, 14484, 14491, 14494, 14499, 14529, 14576, 14625, 14676, 14729, 14784, 14841, 14844, 14849, 14900, 14916, 14925, 14936, 14949, 14961, 14964, 14981, 15291, 15294, 15299, 15761, 15764, 15769, 16116, 16125, 16136, 16149, 16161, 16164, 16169, 16181, 16251, 16254, 16259, 16361, 16364, 16369, 16416, 16425, 16436, 16449, 16464, 16481, 16491, 16494, 16499, 16641, 16644, 16649, 16761, 16764, 16769, 16811, 16814, 16819, 16911, 16914, 16916, 16919, 16925, 16936, 16941, 16944, 16949, 16964, 16981, 16991, 16994, 16999, 17291, 17294, 17299, 17841, 17844, 17849, 18116, 18125, 18136, 18149, 18164, 18181, 18411, 18414, 18419, 19001, 19004, 19009, 19100, 19121, 19144, 19169, 19196, 19225, 19256, 19289, 19324, 19361, 19400, 19441, 19484, 19529, 19576, 19611, 19614, 19619, 19625, 19641, 19644, 19649, 19676, 19691, 19694, 19699, 19729, 19784, 19841, 19900, 19961, 22511, 22514, 22519, 22541, 22544, 22549, 22591, 22594, 22599, 25116, 25125, 25136, 25149, 25161, 25164, 25169, 25181, 25251, 25254, 25259, 25361, 25364, 25369, 25416, 25425, 25436, 25449, 25464, 25481, 25491, 25494, 25499, 25611, 25614, 25619, 25641, 25644, 25649, 25691, 25694, 25699, 25811, 25814, 25819, 25916, 25925, 25936, 25949, 25964, 25981, 28911, 28914, 28919, 28941, 28944, 28949, 28991, 28994, 28999, 32411, 32414, 32419, 32441, 32444, 32449, 32491, 32494, 32499, 36111, 36114, 36116, 36119, 36125, 36136, 36141, 36144, 36149, 36161, 36164, 36169, 36181, 36191, 36194, 36199, 36251, 36254, 36259, 36361, 36364, 36369, 36416, 36425, 36436, 36449, 36464, 36481, 36491, 36494, 36499, 36641, 36644, 36649, 36811, 36814, 36819, 36916, 36925, 36936, 36949, 36964, 36981

Here we see that \(36981= 6^2 \, | \, 3^2 \, | \,9^2\). Once again, we can thin the above numbers by adding the requirement that the number formed by the concatenation be a square number. In this case, we get (permalink):

144, 441, 1369, 1936, 11449, 11664, 14400, 16641, 36481

Here we see that \(36481=6^2 \, | \, 2^2 \, | \,9^2 = 191^2\). None of these sequences of numbers appear in the OEIS as far as I'm aware and I certainly won't be adding them. So nothing of deep mathematical significance in this post, just playing around with powers of numbers and concatenating them. Of course, I've written about Primes Formed By Concatenation quite recently on June 17th 2023.