Xenodrome Pairs, Triplets Etc.

\( \textbf{27815} \), the number associated with my diurnal age today, is one of those numbers for which an interesting property is hard to come by. However, I did notice that the number was a xenodrome in both base 10 and base 9 (42135). A little further investigation revealed that 27816 also shared this property. That got me thinking about how often pairs of such numbers occur. Now I've written about xendromes before in posts titled Xenodromes and Xenodrome Probabilities in which I've looked at numbers that remain xenodromes in various bases but so far I've not looked at groupings of numbers by pairs, triplets, quadruplets etc. with properties relating to xenodromes.

In my Bespoken for Sequences database, I've listed the following sequence of numbers:$$ \begin{align} \textbf{Smaller of a pair of consecutive numbers} \\ \textbf{that are xenodromes in base 10 and base 9} \end{align} $$Between 27815 and 40000, there are 282 numbers with this property (permalink):

27815, 27834, 27835, 27845, 27860, 27950, 27953, 27960, 28013, 28016, 28134, 28169, 28195, 28196, 28314, 28346, 28356, 28364, 28395, 28456, 28495, 28509, 28536, 28563, 28573, 28590, 28609, 28670, 28914, 28934, 28935, 28963, 29015, 29016, 29053, 29075, 29084, 29085, 29103, 29134, 29147, 30147, 30148, 30156, 30157, 30186, 30194, 30195, 30196, 30457, 30458, 30467, 30528, 30548, 30561, 30591, 30691, 30724, 30725, 30751, 30764, 30814, 30825, 30851, 30924, 30925, 31024, 31025, 31048, 31057, 31086, 31094, 31095, 31096, 31097, 31257, 31258, 31259, 31267, 31268, 31284, 31294, 31475, 31586, 31604, 31607, 31608, 31806, 31824, 31825, 31826, 31859, 31905, 31906, 31907, 32018, 32104, 32108, 32159, 32189, 32496, 32508, 32509, 32546, 32580, 32589, 32590, 32607, 32608, 32609, 32648, 32657, 32670, 32689, 32690, 32697, 32907, 32947, 32960, 32964, 34027, 34058, 34085, 34086, 34091, 34095, 34127, 34157, 34158, 34175, 34185, 34206, 34215, 34275, 34278, 34279, 34296, 34297, 34560, 34567, 34568, 34569, 34578, 34620, 34650, 34658, 34761, 34785, 34815, 34820, 34905, 34917, 34926, 35016, 35017, 35047, 35097, 35169, 35196, 35197, 35208, 35216, 35217, 35479, 35486, 35496, 35641, 35809, 35826, 35890, 35891, 35916, 35917, 35946, 35961, 35970, 35971, 35980, 35981, 36018, 36027, 36028, 36208, 36209, 36214, 36218, 36270, 36278, 36279, 36280, 36289, 36290, 37195, 37204, 37245, 37285, 37294, 37295, 37458, 37459, 37485, 37495, 37519, 37520, 37528, 37568, 37819, 37820, 37845, 37864, 37890, 37891, 37920, 37945, 37964, 37981, 38015, 38024, 38045, 38046, 38064, 38105, 38106, 38124, 38145, 38159, 38169, 38205, 38206, 38240, 38249, 38250, 38259, 38260, 38405, 38406, 38415, 38420, 38469, 38649, 38670, 38674, 38694, 38701, 38720, 38724, 38751, 38760, 38904, 38916, 39015, 39016, 39017, 39024, 39025, 39026, 39124, 39125, 39126, 39186, 39205, 39206, 39207, 39215, 39216, 39240, 39260, 39481, 39485, 39540, 39541, 39571, 39580, 39604, 39701, 39715, 39724, 39764, 39805, 39814, 39845, 39846

There are 69  triplets in the same range (permalink):$$ \begin{align} \textbf{Smallest of a triplet of consecutive numbers} \\ \textbf{that are xenodromes in base 10 and base 9} \end{align} $$27834, 28195, 28934, 29015, 29084, 30147, 30156, 30194, 30195, 30457, 30724, 30924, 31024, 31094, 31095, 31096, 31257, 31258, 31267, 31607, 31824, 31825, 31905, 31906, 32508, 32589, 32607, 32608, 32689, 34085, 34157, 34278, 34296, 34567, 34568, 35016, 35196, 35216, 35890, 35916, 35970, 35980, 36027, 36208, 36278, 36279, 36289, 37294, 37458, 37519, 37819, 37890, 38045, 38105, 38205, 38249, 38259, 38405, 39015, 39016, 39024, 39025, 39124, 39125, 39205, 39206, 39215, 39540, 39845

An example is the xenodrome 27834 with 27835 and 27836 also xenodromes. The base 9 equivalents 42156, 42157 and 42158 are also xenodromes.

There are 13 quadruplets in the same range (permalink):$$ \begin{align} \textbf{Smallest of a quaduplet of consecutive numbers} \\ \textbf{that are xenodromes in base 10 and base 9} \end{align} $$30194, 31094, 31095, 31257, 31824, 31905, 32607, 34567, 36278, 39015, 39024, 39124, 39205

An example is the xenodrome 30194 with 30195, 30196 and 30197 also xenodromes. The base 9 equivalents 45368, 45370, 45371 and 45372 are also xenodromes. 

There is only one \( \textbf{quintuplet}\) in the range and that is 31094. Here we see that 31094, 31095, 31096, 31097 and 31098 are all xenodromes as are their base 9 equivalents 46578, 46580, 46581, 46582 and 46583.