A116997 Length of chain starting 2n, iterating f(m) = m - (number of distinct representations of m as the sum of two primes).

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

Offset

2,1

Comments

We start at n=2 as the Goldbach Conjecture is for even integers starting from 4. There is no upper bound for a(n).

Links

Table of n, a(n) for n=2..88.

Formula

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).

Example

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).

Crossrefs

Cf. A000040, A005843, A045917.

Sequence in context: A025128 A058769 A194312 * A050142 A092964 A183368

Adjacent sequences: A116994 A116995 A116996 * A116998 A116999 A117000

Keyword

easy,nonn

Author

Jonathan Vos Post, Apr 03 2006

Extensions

More terms from Giovanni Resta, Jun 15 2016

Status

approved