

A217403


a(n) is the smallest possible largest prime factor of the difference of two prime numbers q > p such that p + q = 2n.


1



2, 2, 2, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 5, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 5, 3, 3, 2, 3, 3, 5, 3, 3, 2, 5, 3, 2, 3, 3, 2, 3, 3, 5, 2, 5, 2, 3, 3, 2, 3, 3, 2, 3, 3, 5, 3, 3, 2, 5, 3, 5, 3, 3, 2, 3, 3, 7
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OFFSET

4,1


COMMENTS

Test up to n=1000000 shows that,
when n is odd, a(n)<=prime(m+3) such that m is the number of prime factors of n that are smaller or equal to prime(m+3);
when n is even, a(n)<=prime(m+4) such that m is the number of prime factors of n that are smaller or equal to prime(m+4).
This is hypothesized true for all n >= 4.
The first appearance of prime(k) in this sequence is A217016.


LINKS

Lei Zhou, Table of n, a(n) for n = 4..10000


EXAMPLE

For n<4, there is not a pair of different prime numbers such that p+q=2n.
When n=4, we have p=3 and q=5 such that 3+5=2n=8 (the only case). The largest prime factor of qp=2 is 2, so a(4)=2;
...
When n=90, we have prime number pairs (7,173) (13,167), (17,163), (23,157), (29,151), (31,149), (41,139), (43,137), (53,127), (67,113), (71,109), (73,107), (79,101), (83,97), totalled 14 pairs such that p1+p2=2n=180. The difference of each pairs are 1737=166=2*83, 16317=154=2*7*11, 16317=146=2*73, 15723=134=2*67, 15129=122=2*61, 14931=118=2*59, 13941=98=2*7^2, 13743=94=2*47, 12753=74=2*37, 11367=46=2*23, 10971=38=2*19, 10773=34=2*17, 10179=22=2*11, and 9783=14=2*7 respectively. Among these prime factorizations, the largest prime factors for each pairs are, 83, 11, 73, 67, 61, 59, 7, 47, 37, 23, 19, 17, 11, and 7 respectively. In the fourteen prime numbers, the smallest one is 7. So a(90)=7.
Additionally, 90=2*3^2*5, prime(3)=5, a(90)=7<prime(3+4)=17, consistent with the hypothesis.


MATHEMATICA

Table[a = i; Do[If[PrimeQ[i  j] && PrimeQ[i + j], f = Last[FactorInteger[j*2]][[1]]; If[a > f, a = f]], {j, 1, i  1}]; a, {i, 4, 100}]


CROSSREFS

Cf. A002375, A217016.
Sequence in context: A199800 A165035 A236531 * A081309 A010553 A262095
Adjacent sequences: A217400 A217401 A217402 * A217404 A217405 A217406


KEYWORD

nonn,easy


AUTHOR

Lei Zhou, Feb 18 2013


STATUS

approved



