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The NASCAR Prime Puzzle
There are 40 possible two-digit endings of primes (with leading zeros). Consider the following "race" after the first 1000 primes:
Let a(n) = the number of primes that end in "01" among the first 10^n primes.
The sequence begins 0, 2, 25, 254, 2494, 24959, 249814, 2499088, 24998779, ...

Let a(n) = the number of primes that end in "03" among the first 10^n primes.
The sequence begins 0, 2, 25, 249, 2510, 25056, 250276, 2500054, 24998487, ...

Let a(n) = the number of primes that end in "07" among the first 10^n primes.
The sequence begins 0, 2, 27, 249, 2459, 24931, 250103, 2500735, 25000294, ...

Let a(n) = the number of primes that end in "09" among the first 10^n primes.
The sequence begins 0, 3, 24, 245, 2504, 24961, 249670, 2500222, 25001398, ...

Let a(n) = the number of primes that end in "11" among the first 10^n primes.
The sequence begins 1, 3, 25, 257, 2492, 25048, 249864, 2499701, 25001011, 

Let a(n) = the number of primes that end in "13" among the first 10^n primes.
The sequence begins 1, 3, 23, 256, 2489, 24956, 249883, 2499909, 25002129, ...

Let a(n) = the number of primes that end in "17" among the first 10^n primes.
The sequence begins 1, 2, 25, 253, 2519, 25001, 250172, 2500991, 24998892, ...

Let a(n) = the number of primes that end in "19" among the first 10^n primes.
The sequence begins 1, 2, 27, 248, 2514, 25003, 250137, 2499557, 24999197, ...

Let a(n) = the number of primes that end in "21" among the first 10^n primes.
The sequence begins 0, 2, 27, 250, 2486, 24973, 249850, 2499065, 24999554, ...

Let a(n) = the number of primes that end in "23" among the first 10^n primes. 
The sequence begins 1, 3, 28, 259, 2511, 25012, 249966, 2499856, 24999237, ...

Let a(n) = the number of primes that end in "27" among the first 10^n primes. 
The sequence begins 0, 2, 23, 250, 2504, 24931, 250074, 2499704, 24999642, ...

Let a(n) = the number of primes that end in "29" among the first 10^n primes.
The sequence begins 1, 2, 26, 255, 2491, 24966, 249873, 2499819, 24999296, ...

Let a(n) = the number of primes that end in "31" among the first 10^n primes.
The sequence begins 0, 4, 24, 251, 2502, 24973, 249851, 2499874, 25001020, ...

Let a(n) = the number of primes that end in "33" among the first 10^n primes.
The sequence begins 0, 2, 24, 247, 2509, 24981, 250124, 2499684, 24999545, ...

Let a(n) = the number of primes that end in "37" among the first 10^n primes. 
The sequence begins 0, 3, 25, 254, 2522, 24959, 249813, 2499850, 25001924, ...

Let a(n) = the number of primes that end in "39" among the first 10^n primes. 
The sequence begins 0, 3, 23, 254, 2512, 25041, 249731, 2499788, 25001154, ...

Let a(n) = the number of primes that end in "41" among the first 10^n primes.
The sequence begins 0, 3, 21, 237, 2505, 24960, 249754, 2500494, 249998836, ...

Let a(n) = the number of primes that end in "43" among the first 10^n primes.
The sequence begins 0, 2, 24, 246, 2484, 25006, 249884, 2500043, 25002072, ...

Let a(n) = the number of primes that end in "47" among the first 10^n primes.
The sequence begins 0, 2, 26, 248, 2520, 24992, 249765, 2499987, 25002877, ...

Let a(n) = the number of primes that end in "49" among the first 10^n primes.
The sequence begins 0, 3, 23, 245, 2516, 24980, 249954, 2499220, 25000053, ...

Let a(n) = the number of primes that end in "51" among the first 10^n primes.
The sequence begins 0, 2, 28, 253, 2504, 25015, 250178, 2500410, 25001559, ...

Let a(n) = the number of primes that end in "53" among the first 10^n primes.
The sequence begins 0, 2, 27, 250, 2497, 24974, 250005, 2499972, 24999702, ...

Let a(n) = the number of primes that end in "57" among the first 10^n primes.
The sequence begins 0, 3, 29, 246, 2528, 25040, 250114, 2499937, 24999120, ...

Let a(n) = the number of primes that end in "59" among the first 10^n primes.
The sequence begins 0, 2, 26, 254, 2512, 25007, 250264, 2500225, 24999949, ...

Let a(n) = the number of primes that end in "61" among the first 10^n primes.
The sequence begins 0, 2, 23, 239, 2491, 24951, 250117, 2500005, 24998609, ...

Let a(n) = the number of primes that end in "63" among the first 10^n primes.
The sequence begins 0, 3, 23, 246, 2516, 25021, 250080, 2500156, 25000087, ...

Let a(n) = the number of primes that end in "67" among the first 10^n primes.
The sequence begins 0, 4, 25, 250, 2479, 25038, 250019, 2499172, 24999835, ...

Let a(n) = the number of primes that end in "69" among the first 10^n primes.
The sequence begins 0, 1, 22, 247, 2479, 25021, 249895, 2500196, 24999156, ...

Let a(n) = the number of primes that end in "71" among the first 10^n primes.
The sequence begins 0, 2, 23, 245, 2500, 24995, 250012, 2500832, 24999429, ...

Let a(n) = the number of primes that end in "73" among the first 10^n primes.
The sequence begins 0, 3, 27, 244, 2474, 25040, 249980, 2500778, 25001960, ...

Let a(n) = the number of primes that end in "77" among the first 10^n primes.
The sequence begins 0, 1, 25, 242, 2488, 25099, 250165, 2499766, 25000007, ...

Let a(n) = the number of primes that end in "79" among the first 10^n primes.
The sequence begins 0, 4, 26, 253, 2503, 25059, 250054, 2500497, 24999733, ...

Let a(n) = the number of primes that end in "81" among the first 10^n primes.
The sequence begins 0, 2, 24, 251, 2487, 25001, 249942, 2500072, 25000149, ...

Let a(n) = the number of primes that end in "83" among the first 10^n primes.
The sequence begins 0, 3, 27, 256, 2510, 24978, 249909, 2499642, 25000408, ...

Let a(n) = the number of primes that end in "87" among the first 10^n primes.
The sequence begins 0, 1, 23, 255, 2516, 25046, 249966, 2500611, 24999550, ...

Let a(n) = the number of primes that end in "89" among the first 10^n primes.
The sequence begins 0, 2, 25, 247, 2484, 24914, 250134, 2499991, 24998766, ...

Let a(n) = the number of primes that end in "91" among the first 10^n primes.
The sequence begins 0, 2, 25, 247, 2506, 25059, 250373, 2499894, 24998633, ...

Let a(n) = the number of primes that end in "93" among the first 10^n primes.
The sequence begins 0, 2, 24, 261, 2506, 25085, 250101, 2500040, 24998592, ...

Let a(n) = the number of primes that end in "97" among the first 10^n primes.
The sequence begins 0, 3, 25, 260, 2479, 24976, 250091, 2499648, 25000363, ...

Let a(n) = the number of primes that end in "99" among the first 10^n primes.
The sequence begins 0, 2, 24, 243, 2494, 24988, 250039, 2500511, 24998992, ...
... for example "Car Number 57" is leading after the first 10^4 laps (primes). But what about after 10^5 laps? 10^6 laps, etc.? Place your bets now!
Update: Thanks to Chuck Gaydos of Arizona, we now know that "Car Number 47" is in the lead with 25002877 primes after 10^9 (one billion) laps.

*Sequences and puzzle proposed by G. L. Honaker, Jr. (June 2016)

 

  Submitted: 2016-07-22 20:49:02;   Last Modified: 2018-04-06 16:51:35.



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