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A156695 Odd numbers that are not of the form p + 2^a + 2^b, a, b > 0, p prime. 47
1, 3, 5, 6495105, 848629545, 1117175145, 2544265305, 3147056235, 3366991695, 3472109835, 3621922845, 3861518805, 4447794915, 4848148485, 5415281745, 5693877405, 6804302445, 7525056375, 7602256605, 9055691835, 9217432215 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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).

EXAMPLE

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

PROG

(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

CROSSREFS

Cf. A006285, A118955, A232565.

Sequence in context: A319598 A126334 A068635 * A175645 A178514 A154924

Adjacent sequences:  A156692 A156693 A156694 * A156696 A156697 A156698

KEYWORD

nonn,hard,nice

AUTHOR

Charles R Greathouse IV, Feb 13 2009

EXTENSIONS

Factorizations added by Daniel Forgues, Jan 20 2011

STATUS

approved

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Last modified May 26 09:27 EDT 2019. Contains 323579 sequences. (Running on oeis4.)