cp's OEIS Frontend

Is there some absolute upper limit of k for each n, after which the program can finish the testing loop? - Antti Karttunen, Dec 20 2009

Reply from T. D. Noe, Dec 20 2009: Although theorem 2.1 in the paper by Stolarsky is useful, the seqfan e-mail from Jack Brennen sometime around July 2008 is the key to computing these numbers. "To determine if an odd number N is flimsy, take the finite set of residues of 2^a (mod N). Assume that the number of 1's in the binary representation of N is equal to C. To show that the number is flimsy, find a way to construct zero (mod N) by adding up some number of residues of 2^a (mod N) using less than C terms. To show that the number is sturdy, show that it's impossible to do so." In short, this sequence, though difficult to compute, is well defined.

Numbers of the form 2^m-1 (A000225) is a subsequence. - David A. Corneth, Oct 01 2016

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.

From T. D. Noe, Mar 01 2010: (Start)

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

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)

If n is a power of 2 then a(n)=1. All other positive n have a(n)>1. a(n)=2 precisely in cases where some multiple of n is a factor of 2^q+1 for some q.

a(n)=0 indicates that n is a sturdy number (A125121); that is, no multiple of n has fewer ones than the binary expansion of n. If a(n)>0, n is a flimsy number (A005360). Compare with A143069.

Or, binary weight of 5*k is less than binary weight of k.

Numbers k such that A000120(k) > A000120(5*k).

Also called the 5-flimsy numbers; see the Stolarsky reference.

If m is here, 2m is too. Hence the "primitive solutions" are all odd ones: 13,29,53,55,61,77,103,109,111,117,119,125,157,205,207,213,215,219,221,223,231, ...