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2, 3, 5, 7, 17, 31, 73, 89, 127, 257, 1801, 2089, 8191, 65537, 131071, 178481, 262657, 524287, 2099863, 616318177, 2147483647, 4432676798593
From Jason Yuen, Mar 30 2024: (Start)
For all x>log_2(p), 1+A000120(p-(2^x mod p)) >= A000120(p). This follows from the fact that 2^x+p-(2^x mod p) is a multiple of p.
a(23) > 5*10^12. See a143027_5e12.txt for more details. (End)
Jason Yuen, <a href="/A143027/a143027.txt">a143027_5e12.txt</a>. This file shows that a(23) > 5*10^12.
4432676798593 added by Jason Yuen, Mar 30 2024
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Sturdy prime numbers: p such that in binary notation k*p has at least as many 1-bits as p for all k > 0.
The primes in A125121. This sequence includes the Fermat primes (A019434), Mersenne primes (A000668) and the three known primes in A051154, . It appears that almost all primes are flimsy numbers, A005360.
Clokie et al. verify that the next two sturdy primes after 2099863 are 616318177 and 2147483647. These are all up to 2^{32}. Two additional sturdy primes are 57912614113275649087721 = (2^{83} - 1)/167 and 10350794431055162386718619237468234569 = (2^{131} - 1)/263, but probably there are some in between these and 2147483647. Jeffrey Shallit, Feb 10 2020
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