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:

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L=[]

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for n in [10..40000]:

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    s=str(n*prime_pi(n))

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    count=1

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    for i in range(len(s)-1):

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        if s[i]!=s[i+1]:

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            count-=1

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            if count==0:

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                break

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        else:

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            count=2

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    if count>1:

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        L.append(n)

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print(L)

Language: Sage Gap GP HTML Macaulay2 Maxima Octave Python R Singular

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Messages

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.

xxxxxxxxxx

 

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L=[]

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L=[]

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for n in [10..40000]:

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    s=str(n)

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    count=1

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    for i in range(len(s)-1):

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        if s[i]!=s[i+1]:

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            count-=1

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            if count==0:

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                break

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        else:

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            count=2

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    if count>1:

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        L.append(n)

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print(L)

Language: Sage Gap GP HTML Macaulay2 Maxima Octave Python R Singular

---

Messages

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:

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