A156695 Odd numbers that are not of the form p + 2^a + 2^b, a, b > 0, p prime.

1, 3, 5, 6495105, 848629545, 1117175145, 2544265305, 3147056235, 3366991695, 3472109835, 3621922845, 3861518805, 4447794915, 4848148485, 5415281745, 5693877405, 6804302445, 7525056375, 7602256605, 9055691835, 9217432215

Offset: 1

Author

Charles R Greathouse IV, Feb 13 2009

Keywords

• nonn
• hard
• nice

Comments

Crocker shows that this sequence is infinite.

All members above 5 found so far (up to 2.5 * 10^11) are divisible by 255 = 3 * 5 * 17, and many are divisible by 257. I conjecture that all members of this sequence greater than 5 are divisible by 255. This implies that all odd numbers (greater than 7) are the sum of a prime and at most three positive powers of two.

Pan shows that, for every c > 1, a(n) << x^c. More specifically, there are constants C,D > 0 such that there are at least Dx/exp(C log x log log log log x/log log log x) members of this sequence up to x. - Charles R Greathouse IV, Apr 11 2016

All terms > 5 are numbers k > 3 such that k - 2^n is a de Polignac number (A006285) for every n > 0 with 2^n < k. Are there numbers K such that |K - 2^n| is a Riesel number (A101036) for every n > 0? If so, ||K - 2^n| - 2^m| is composite for every pair m,n > 0, by the dual Riesel conjecture. - Thomas Ordowski, Jan 06 2024

In keeping with the example's connection to A000215, the lowest ki for ki * Product_{i=0..11} (F(i)) to belong to A156695 are 1, 433007, 25471, 17047, 1291, 7, 101, 807, 83, 347, 9, 179. So for example, 433007*(3*5) is a term. This implies a variant of the first commented conjecture accordingly. - Bill McEachen, Apr 17 2025

Examples

			Prime factorization of terms:
F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257 are Fermat numbers (cf. A000215)
6495105    = 3   * 5   * 17               * 25471
848629545  = 3   * 5   * 17               * 461      * 7219
1117175145 = 3   * 5   * 17         * 257 * 17047
2544265305 = 3^2 * 5   * 17         * 257 * 12941
3147056235 = 3^2 * 5   * 17         * 257 * 16007
3366991695 = 3   * 5   * 17   * 83  * 257 * 619
3472109835 = 3   * 5   * 17         * 257 * 52981
3621922845 = 3   * 5   * 17^2       * 257 * 3251
3861518805 = 3^3 * 5   * 17         * 257 * 6547
4447794915 = 3^3 * 5   * 17         * 257 * 7541
4848148485 = 3^4 * 5   * 17               * 704161
5415281745 = 3   * 5   * 17               * 21236399
5693877405 = 3^2 * 5   * 17         * 257 * 28961
6804302445 = 3^2 * 5   * 17   * 53  * 257 * 653
7525056375 = 3^2 * 5^3 * 17         * 257 * 1531
7602256605 = 3   * 5   * 17         * 257 * 311      * 373
9055691835 = 3   * 5   * 17         * 257 * 138181
9217432215 = 3^2 * 5   * 17   * 173 * 257 * 271
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..233 (terms < 10^12)
• Roger Crocker, "On the sum of a prime and of two powers of two", Pacific Journal of Mathematics 36:1 (1971), pp. 103-107.
• Roger Crocker, Some counter-examples in the additive theory of numbers, Master's thesis (Ohio State University), 1962.
• Hao Pan, On the integers not of the form p + 2^a + 2^b. arXiv:0905.3809 [math.NT], 2009.
• Zhi-Wei Sun, Mixed sums of primes and other terms (2009-2010).

Crossrefs

Cf. A006285, A118955, A232565, A337487.

Programs

PARI

is(n)=if(n%2==0,return(0)); for(a=1,log(n)\log(2), for(b=1,a, if(isprime(n-2^a-2^b),return(0)))); 1 \\ Charles R Greathouse IV, Nov 27 2013

Python

from itertools import count, islice
from sympy import isprime
def A156695_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue+(startvalue&1^1),1),2):
        l = n.bit_length()-1
        for a in range(l,0,-1):
            c = n-(1<A156695_list = list(islice(A156695_gen(),4)) # Chai Wah Wu, Nov 29 2023

Extensions

Factorizations added by Daniel Forgues, Jan 20 2011