Sunday, 4 December 2016

Lucky Numbers

I haven't devoted a entire post to lucky numbers before, even though I have mentioned them (Prime Number Chains). They occur with about the same frequency as prime numbers but whereas WolframAlpha makes a note of prime numbers, it does not mention lucky numbers. So far they just pass by unnoticed. For the sake of completeness, let's define a lucky number once again (taken from WolframAlpha):
Write out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first odd number >1 is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, .... The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, ....  
Numbers remaining after this procedure has been carried out completely are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ... (OEIS A000959). Many asymptotic properties of the prime numbers are shared by the lucky numbers. The asymptotic density is 1/lnN, just as the prime number theorem, and the frequency of twin primes and twin lucky numbers are similar. A version of the Goldbach conjecture also seems to hold.
The OEIS site also has a list of lucky numbers between 1 and 200000. I've extracted a list of those that are coming up for me:


2495th lucky number: 24727
2496th lucky number: 24729
2497th lucky number: 24733 lucky and prime
2498th lucky number: 24739
2499th lucky number: 24741
2500th lucky number: 24759
2501st lucky number: 24763 lucky and prime
2502nd lucky number: 24771
2503rd lucky number: 24783
2504th lucky number: 24789
2505th lucky number: 24805
2506th lucky number: 24811
2507th lucky number: 24829
2508th lucky number: 24831
2509th lucky number: 24843
2510th lucky number: 24855
2511th lucky number: 24865
2512th lucky number: 24873
2513th lucky number: 24877 lucky and prime
2514th lucky number: 24895
2515th lucky number: 24907 lucky and prime
2516th lucky number: 24909
2517th lucky number: 24933
2518th lucky number: 24951
2519th lucky number: 24957
2520th lucky number: 24963
2521st lucky number: 24985
2522nd lucky number: 24991

I'll begin entering these into my calendar so that I'm reminded on what numbers are lucky as they occur.

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