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.
Odd sturdy primes appear to be the largest primitive prime factor of 2^q-1 for q a prime or prime power. The values of q for the current terms: 2, 4, 3, 8, 5, 9, 11, 16, 25, 29, 13, 32, 17, 23, 27 and 19. The sequence probably continues with 2099863, 6700417, 13264529, 20394401, 97685839.
Max Alekseyev reports that 6700417, 13264529, 20394401, and 97685839 are not sturdy because each number divides a number having fewer 1-bits: 6700417 divides 2^32 + 1, 13264529 divides 331613225, 20394401 divides 1611157679, and 97685839 divides 18014398643699713. He conjectures that 616318177 is the next term.
If q is a prime power, q = r^s, then the primitive part of 2^q-1 is (2^r^s-1)/(2^r^(s-1)-1). According to Stolarsky's Theorem 2.1, this primitive part is sturdy. If the primitive part is prime, then it is in this sequence. Hence 7^2 produces the sturdy prime 4432676798593 and 59^2 produces a 1031-digit sturdy prime. (End)
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
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)
