Doublets, Triplets etc.

One of the many properties associated with my diurnal age today of 26553 is that it's a member of OEIS A116057: 

A116057

\(n\) times \( \Pi(n) \) is made of nontrivial runs of identical digits, where \( \Pi(n) \) is the prime counting function that returns the number of primes less than or equal to a given number.

It took me a little thought to get an algorithm that would return the members of this sequence but eventually I succeeded. While it may not be the most elegant of algorithms, it does get the job done. I've embedded the code from SageMathCell (permalink) below:

As can be seen the initial members of the sequence, up to 40,000, are:

[11, 37, 66, 154, 332, 750, 1696, 4000, 13684, 22308, 26640, 26653, 30327]

Taking the last member of the sequence, it can be seen that:$$ 30377 \times { \Large \Pi } (30327) =99442233 $$The algorithm is easily modifiable and below it is used to generate the initial terms of OEIS A033023:

A033023

Numbers whose base-10 expansion has no run of digits with length < 2.

Here the members of sequence, up to 40,000, are:

[11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1100, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 2200, 2211, 2222, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 3300, 3311, 3322, 3333, 3344, 3355, 3366, 3377, 3388, 3399, 4400, 4411, 4422, 4433, 4444, 4455, 4466, 4477, 4488, 4499, 5500, 5511, 5522, 5533, 5544, 5555, 5566, 5577, 5588, 5599, 6600, 6611, 6622, 6633, 6644, 6655, 6666, 6677, 6688, 6699, 7700, 7711, 7722, 7733, 7744, 7755, 7766, 7777, 7788, 7799, 8800, 8811, 8822, 8833, 8844, 8855, 8866, 8877, 8888, 8899, 9900, 9911, 9922, 9933, 9944, 9955, 9966, 9977, 9988, 9999, 11000, 11100, 11111, 11122, 11133, 11144, 11155, 11166, 11177, 11188, 11199, 11222, 11333, 11444, 11555, 11666, 11777, 11888, 11999, 22000, 22111, 22200, 22211, 22222, 22233, 22244, 22255, 22266, 22277, 22288, 22299, 22333, 22444, 22555, 22666, 22777, 22888, 22999, 33000, 33111, 33222, 33300, 33311, 33322, 33333, 33344, 33355, 33366, 33377, 33388, 33399, 33444, 33555, 33666, 33777, 33888, 33999]

Of course, the algorithm is easily modifiable to find numbers whose base-\(n\) expansion has no run of digits with length < 2. Here \(n \) can range from 2 to 36. For example, if \(n\)=16, then the numbers up to 40,000 that satisfy are:

[17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 273, 546, 819, 1092, 1365, 1638, 1911, 2184, 2457, 2730, 3003, 3276, 3549, 3822, 4095, 4352, 4369, 4386, 4403, 4420, 4437, 4454, 4471, 4488, 4505, 4522, 4539, 4556, 4573, 4590, 4607, 8704, 8721, 8738, 8755, 8772, 8789, 8806, 8823, 8840, 8857, 8874, 8891, 8908, 8925, 8942, 8959, 13056, 13073, 13090, 13107, 13124, 13141, 13158, 13175, 13192, 13209, 13226, 13243, 13260, 13277, 13294, 13311, 17408, 17425, 17442, 17459, 17476, 17493, 17510, 17527, 17544, 17561, 17578, 17595, 17612, 17629, 17646, 17663, 21760, 21777, 21794, 21811, 21828, 21845, 21862, 21879, 21896, 21913, 21930, 21947, 21964, 21981, 21998, 22015, 26112, 26129, 26146, 26163, 26180, 26197, 26214, 26231, 26248, 26265, 26282, 26299, 26316, 26333, 26350, 26367, 30464, 30481, 30498, 30515, 30532, 30549, 30566, 30583, 30600, 30617, 30634, 30651, 30668, 30685, 30702, 30719, 34816, 34833, 34850, 34867, 34884, 34901, 34918, 34935, 34952, 34969, 34986, 35003, 35020, 35037, 35054, 35071, 39168, 39185, 39202, 39219, 39236, 39253, 39270, 39287, 39304, 39321, 39338, 39355, 39372, 39389, 39406, 39423]

In the OEIS, this is sequence A033029:

A033029

Numbers whose base-16 expansion has no run of digits with length < 2.

Here is a permalink to the algorithm. For example, take 26316 from this list as an example. We have \(26316_{{\small 10}}=66\text{cc}_{ {\small 16}}\).

The algorithm can be used to find some numbers with interesting properties. For example, in the range up to one million, what members of OEIS A033023 (sequence listed earlier) have a totient that is also a member of OEIS A033023? I won't embed the code but here is a permalink to the algorithm that I used to find the numbers with the required property. It turns out that there are only three:

• \( \phi(9922)=4400\)
• \( \phi(662233)=554400\)
• \( \phi(990022)=449900\)