%I #36 Apr 20 2024 10:01:30

%S 2,3,5,7,17,31,73,89,127,257,1801,2089,8191,65537,131071,178481,

%T 262657,524287,2099863,616318177,2147483647,4432676798593

%N Sturdy prime numbers: p such that in binary notation k*p has at least as many 1-bits as p for all k > 0.

%C 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.

%C 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.

%C From _T. D. Noe_, Mar 01 2010: (Start)

%C _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.

%C 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)

%C 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

%C From _Jason Yuen_, Mar 30 2024: (Start)

%C 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.

%C a(23) > 5*10^12. See a143027_5e12.txt for more details. (End)

%H Trevor Clokie et al., <a href="https://arxiv.org/abs/2002.02731">Computational Aspects of Sturdy and Flimsy Numbers</a>, arxiv preprint arXiv:2002.02731 [cs.DS], February 7 2020.

%H K. B. Stolarsky, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa38/aa3825.pdf">Integers whose multiples have anomalous digital frequencies,</a> Acta Arithmetica, 38 (1980), 117-128.

%H Jason Yuen, <a href="/A143027/a143027.txt">a143027_5e12.txt</a>. This file shows that a(23) > 5*10^12.

%Y Cf. A125121, A181863, A000120.

%K more,nice,nonn,base

%O 1,1

%A _T. D. Noe_, Jul 17 2008, Jul 21 2008

%E 2089 and 8191 were found by _Ray Chandler_

%E 2099863 added by _T. D. Noe_, Mar 01 2010

%E 616318177, 2147483647 added by _Jeffrey Shallit_, Feb 10 2020

%E 4432676798593 added by _Jason Yuen_, Mar 30 2024