A181863 -id:A181863 - cp's OEIS frontend

A143027 Sturdy prime numbers: p such that in binary notation k*p has at least as many 1-bits as p for all k > 0.

2, 3, 5, 7, 17, 31, 73, 89, 127, 257, 1801, 2089, 8191, 65537, 131071, 178481, 262657, 524287, 2099863, 616318177, 2147483647, 4432676798593

Offset: 1

T. D. Noe, Jul 17 2008, Jul 21 2008

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)

Trevor Clokie et al., Computational Aspects of Sturdy and Flimsy Numbers, arxiv preprint arXiv:2002.02731 [cs.DS], February 7 2020.
K. B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arithmetica, 38 (1980), 117-128.
Jason Yuen, a143027_5e12.txt. This file shows that a(23) > 5*10^12.

Cf. A125121, A181863, A000120.

2089 and 8191 were found by Ray Chandler
2099863 added by T. D. Noe, Mar 01 2010
616318177, 2147483647 added by Jeffrey Shallit, Feb 10 2020
4432676798593 added by Jason Yuen, Mar 30 2024

A181862 Decimal sturdy numbers: positive integers m such that sum of digits of k * m for any positive integer k is at least the sum of digits of m.

Original entry on oeis.org

1, 3, 9, 10, 11, 12, 18, 21, 27, 30, 33, 36, 41, 45, 54, 63, 72, 81, 90, 99, 100, 101, 102, 108, 110, 111, 117, 120, 123, 126, 132, 135, 144, 153, 162, 171, 180, 198, 201, 207, 210, 216, 225, 231, 234, 243, 252, 261, 270, 297, 300, 303, 306, 315, 324, 330, 333, 342, 351, 360, 396, 405, 410

Offset: 1

Max Alekseyev, Nov 14 2010

Positive integers m such that A007953(m) = A077196(m).
All powers of 10 and many multiples of 3 are in this sequence, many prime numbers are not. Notable exceptions are the primes 11 and 41 that are in this sequence, and multiples of 3 like 6 and 15 that are not.
This suggests that a digit sum of 6 disqualifies a multiple of 3 from this sequence, not parity. A digit sum of 9, by contrast, ensures the number is in this sequence. - Alonso del Arte, Oct 02 2016

			11 has a digit sum of 2. If a multiple of 11 exists with a digit sum of 1, that would mean a power of 10 is also a multiple of 11, which is absurd. Therefore 11 is in the sequence.
12 = 2^2 * 3 has a digit sum of 3. In base 10, all multiples of 3 have a digital root of 3, 6 or 9, which means that a total digit sum of 1 or 2 is impossible for a multiple of 3. Therefore 12 is in the sequence.
13 has a digit sum of 4. However, note that 7 * 11 * 13 = 1001, which has a digit sum of 2. So 13 is not in the sequence.

Jason Yuen, Table of n, a(n) for n = 1..10000
Trevor Clokie, Thomas F. Lidbetter, Antonio Molina Lovett, Jeffrey Shallit, and Leon Witzman, Computational Aspects of Sturdy and Flimsy Numbers, arXiv:2002.02731 [cs.DS], 2020.

Cf. A125121, A181863.

cp's OEIS Frontend

A143027 Sturdy prime numbers: p such that in binary notation k*p has at least as many 1-bits as p for all k > 0.

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