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A116997
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Length of chain starting 2n, iterating f(m) = m - (number of distinct representations of m as the sum of two primes).
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0
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2, 2, 2, 3, 2, 3, 4, 5, 6, 2, 2, 2, 3, 2, 3, 3, 4, 5, 2, 6, 2, 7, 2, 8, 2, 2, 2, 3, 3, 2, 2, 4, 5, 2, 5, 2, 2, 2, 3, 2, 3, 2, 4, 2, 5, 2, 2, 2, 3, 3, 2, 4, 4, 3, 2, 3, 4, 3, 5, 4, 2, 5, 2, 2, 2, 3, 2, 3, 2, 4, 2, 3, 2, 4, 3, 4, 2, 2, 4, 4, 2, 5, 2, 2, 6, 2, 2
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OFFSET
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2,1
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COMMENTS
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We start at n=2 as the Goldbach Conjecture is for even integers starting from 4. There is no upper bound for a(n).
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LINKS
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Table of n, a(n) for n=2..88.
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FORMULA
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a(n) = number of steps k, including start A005843(n) = 2n and end (f^k(n) is odd), where f(2n) = 2n - A045917(n) = 2n - (number of ways of writing 2n as a sum of 2 primes when order does not matter).
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EXAMPLE
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a(2) = 2 because we have two integers in the chain starting 2*2 = 4, since there is 1 unique way to partition 4 into two primes (4=2+2), so f(4) = 4-1 = 3, which is a halting state, f(3)=3, since 3 is odd and not the sum of two primes. The chain is (4,3).
a(3) = 2, the chain being (6,5) since 6 is uniquely 3+3.
a(5) = 3 because 2*5 = 10 = 3+7 = 5+5 (two ways), 10-2 = 8, then 8 is uniquely 3+5 so the chain is (10,f(10),f(f(10))) = (10,8,7) which is of length 3.
a(10) = 6, the chain being (20,18,16,14,12,11).
a(23) = 7, the chain being (46,42,38,36,32,30,27).
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CROSSREFS
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Cf. A000040, A005843, A045917.
Sequence in context: A025128 A058769 A194312 * A050142 A092964 A183368
Adjacent sequences: A116994 A116995 A116996 * A116998 A116999 A117000
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post, Apr 03 2006
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EXTENSIONS
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More terms from Giovanni Resta, Jun 15 2016
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STATUS
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approved
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