A232565 -id:A232565 - cp's OEIS frontend

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

Charles R Greathouse IV, Feb 13 2009

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

			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

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).

Cf. A006285, A118955, A232565, A337487.

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

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

Factorizations added by Daniel Forgues, Jan 20 2011

A268693 Odd numbers that are not of the form p + 2^a + 2^b with b > a > 0, and p prime.

Original entry on oeis.org

1, 3, 5, 7, 255, 15045, 20655, 25755, 39525, 44115, 46665, 65535, 70125, 97155, 100215, 132855, 144465, 162945, 196605, 200175, 277695, 280755, 327675, 441915, 469965, 548505, 688245, 720375, 770355, 773925, 789225, 1017705, 1027395, 1110015, 1114095, 1127355

Offset: 1

Arkadiusz Wesolowski, Feb 11 2016

nonn

This sequence is infinite; in particular, 2^(2^n) - 1 is in this sequence for each n > 2.

Not every member of this sequence greater than 7 is divisible by 255, see A268694.

Kevin P. Thompson, Table of n, a(n) for n = 1..392

Cf. A232565, A268694. Supersequence of A156695.

A369375 Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).

Original entry on oeis.org

1, 7, 15, 23, 27, 31, 37, 39, 43, 55, 58, 63, 71, 79, 82, 91, 95, 111, 123, 127, 133, 135, 139, 143, 148, 151, 159, 167, 169, 172, 173, 175, 179, 183, 191, 195, 199, 207, 211, 223, 239, 255, 286, 295, 313, 316, 319, 335, 337, 351, 367, 373, 383, 406, 415, 417, 433, 435, 447, 455, 461, 463, 479

Offset: 1

Thomas Ordowski, Jan 22 2024

nonn

Integers m > 0 such that 2^m-1 - 2^n is not prime for every natural n < m.
For m > 2, a number m is a term of this sequence if and only if A208083(m) = 0.
All Mersenne number m = 2^k-1 for k > 2 are in this sequence. The proof is below.
Cf. A138290 (see Chai Wah Wu's conjecture in the third comment). By Crocker's (1971) theorem: if m > 2 and a <> b, then 2^(2^m)-1 - 2^a - 2^b is not prime.
If a = 2^m-1, then b < a, so for m > 2, 2^(2^m-1)-1 is a de Polignac number, QED.
Note that 2^(2^m-1)-1 - 2^n is divisible by some prime factor of 2^(2^m)-1.
Prime numbers of this sequence are Mersenne primes > 3, and many other primes.
Conjecture: if n > 5, then |2^(2^n-1)-1 - 2^m| is not prime for every m > 0.
If so, then by the dual Riesel conjecture, 2^(2^n-1)-1 is a (dual) Riesel number, i.e., if n > 5, then (2^(2^n-1)-1)2^m-1 is composite for every integer m > 0.
For example, the double Mersenne prime 2^(2^7-1)-1 may be a dual Riesel number.
It seems that the natural density of these numbers is about twice as high as the density of de Polignac numbers.
For many terms m, 2m+1 is also in this sequence. By iteration (x -> 2x+1), the subsequence b(n) = (m+1)2^n-1, for n >= 0, is infinite if m = 7 (which has already been proven) and probably if m = 27 (which is hard to prove).

			7 is a term since {2^7-1-2, 2^7-1-2^2, 2^7-1-2^3, 2^7-1-2^4, 2^7-1-2^5, 2^7-1-2^6} = {125, 123, 119, 111, 95, 63} and all six of these numbers are composite.
Note that both 2^148-1 and 2^148+1 are de Polignac numbers.

Cf. A000225, A006285, A000668, A101036, A138290, A208083, A232565, A369378.

fQ[n_] := Block[{k = n -1}, While[k > 1 && !PrimeQ[2^n -1 -2^k], k--]; k == 1]; Select[ Range[3, 450], fQ] (* Robert G. Wilson v, Jan 22 2024 *)

For n > 1, a(n) = A138290(n-1) + 1.

A208083(a(n)) = 0, for n > 0.

More terms from Robert G. Wilson v, Jan 22 2024

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

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A268693 Odd numbers that are not of the form p + 2^a + 2^b with b > a > 0, and p prime.

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A369375 Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).

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