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%I #59 Jan 20 2024 09:05:56
%S 1,3,5,6495105,848629545,1117175145,2544265305,3147056235,3366991695,
%T 3472109835,3621922845,3861518805,4447794915,4848148485,5415281745,
%U 5693877405,6804302445,7525056375,7602256605,9055691835,9217432215
%N Odd numbers that are not of the form p + 2^a + 2^b, a, b > 0, p prime.
%C Crocker shows that this sequence is infinite.
%C 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.
%C 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
%C 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
%H Giovanni Resta, <a href="/A156695/b156695.txt">Table of n, a(n) for n = 1..233</a> (terms < 10^12)
%H Roger Crocker, "<a href="http://projecteuclid.org/euclid.pjm/1102971271">On the sum of a prime and of two powers of two</a>", Pacific Journal of Mathematics 36:1 (1971), pp. 103-107.
%H Roger Crocker, <a href="/A156695/a156695.pdf">Some counter-examples in the additive theory of numbers</a>, Master's thesis (Ohio State University), 1962.
%H Hao Pan, <a href="http://arxiv.org/abs/0905.3809">On the integers not of the form p + 2^a + 2^b</a>. arXiv:0905.3809 [math.NT], 2009.
%H Zhi-Wei Sun, <a href="http://math.nju.edu.cn/~zwsun/MSPT.htm">Mixed sums of primes and other terms</a> (2009-2010).
%e Prime factorization of terms:
%e F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257 are Fermat numbers (cf. A000215)
%e 6495105 = 3 * 5 * 17 * 25471
%e 848629545 = 3 * 5 * 17 * 461 * 7219
%e 1117175145 = 3 * 5 * 17 * 257 * 17047
%e 2544265305 = 3^2 * 5 * 17 * 257 * 12941
%e 3147056235 = 3^2 * 5 * 17 * 257 * 16007
%e 3366991695 = 3 * 5 * 17 * 83 * 257 * 619
%e 3472109835 = 3 * 5 * 17 * 257 * 52981
%e 3621922845 = 3 * 5 * 17^2 * 257 * 3251
%e 3861518805 = 3^3 * 5 * 17 * 257 * 6547
%e 4447794915 = 3^3 * 5 * 17 * 257 * 7541
%e 4848148485 = 3^4 * 5 * 17 * 704161
%e 5415281745 = 3 * 5 * 17 * 21236399
%e 5693877405 = 3^2 * 5 * 17 * 257 * 28961
%e 6804302445 = 3^2 * 5 * 17 * 53 * 257 * 653
%e 7525056375 = 3^2 * 5^3 * 17 * 257 * 1531
%e 7602256605 = 3 * 5 * 17 * 257 * 311 * 373
%e 9055691835 = 3 * 5 * 17 * 257 * 138181
%e 9217432215 = 3^2 * 5 * 17 * 173 * 257 * 271
%o (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
%o (Python)
%o from itertools import count, islice
%o from sympy import isprime
%o def A156695_gen(startvalue=1): # generator of terms >= startvalue
%o for n in count(max(startvalue+(startvalue&1^1),1),2):
%o l = n.bit_length()-1
%o for a in range(l,0,-1):
%o c = n-(1<<a)
%o for b in range(min(a,l-1),0,-1):
%o if isprime(c-(1<<b)):
%o break
%o else:
%o continue
%o break
%o else:
%o yield n
%o A156695_list = list(islice(A156695_gen(),4)) # _Chai Wah Wu_, Nov 29 2023
%Y Cf. A006285, A118955, A232565, A337487.
%K nonn,hard,nice
%O 1,2
%A _Charles R Greathouse IV_, Feb 13 2009
%E Factorizations added by _Daniel Forgues_, Jan 20 2011
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