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Search: id:A152451
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| A152451 |
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From every interval (2^(m-1), 2^m), m>=3, we remove primes p for which 2^m-p is a prime; the sequence gives the remaining primes. |
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5
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| 3, 7, 17, 23, 31, 37, 43, 71, 73, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 151, 157, 163, 179, 181, 191, 193, 199, 211, 223, 229, 241, 257, 263, 269, 277, 281, 293, 307, 311, 317, 337, 347, 353, 359, 367, 379, 383, 389, 397, 401, 419, 421, 431, 443
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Powers of 2 are not expressible as sums of two primes from this sequence.
Consider a strong Goldbach conjecture: every even number n>=6 is a sum of two primes the least from which is O((log(n))^2log(log(n))) (sf. comment to A152522). The number of such representations for 2^k, trivially, is less than k^5 for k>k_0. Removing the maximal primes in every of such representations of 2^k, k>=3, we obtain an analog B of A152451 with the counting function H(x)= pi(x)-O((log(x))^5). Replacing in B the first N terms by N consecutive primes (with arbitrary large N), we obtain a sequence which essentially is indistinguishable from the sequence of all primes with help of the approximation of pi(x) by li(x), since, according to well known Littlewood result, the remaider term in the theorem of primes could not be less than sqrt(x)logloglog(x)/log(x). But for this sequence we have infinitely many even numbers for which the considered strong Goldbach conjecture is wrong. Thus the conjecture is essentially unprovable.
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FORMULA
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If A(X) is the counting function for the terms a(n)<=x, then A(x)=x/ln(x)+O(x*ln(ln(x))/(ln(x))^2)
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CROSSREFS
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Sequence in context: A093651 A018411 A083989 this_sequence A097958 A118940 A127175
Adjacent sequences: A152448 A152449 A152450 this_sequence A152452 A152453 A152454
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KEYWORD
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nonn
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AUTHOR
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Vladimir Shevelev (shevelev(AT)bgu.ac.il), Dec 04 2008, Dec 05 2008, Dec 08 2008, Dec 12 2008
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