|
| |
|
|
A045917
|
|
From Goldbach problem: number of decompositions of 2n into unordered sums of two primes.
|
|
42
|
|
|
|
0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3, 7, 9, 6, 5, 8, 7, 8, 11, 6, 5, 12, 4, 8, 11, 5, 8, 10, 5, 6, 13, 9, 6, 11, 7, 7, 14, 6, 8, 13, 5, 8, 11, 7, 9
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,5
|
|
|
COMMENTS
|
Note that A002375 (which differs only at the n=2 term) is the main entry for this sequence.
The graph of this sequence is called Goldbach's comet. [David W. Wilson, Mar 19 2012]
This is the row length sequence of A182138, A184995 and A198292. - Jason Kimberley, Oct 03 2012
|
|
|
REFERENCES
|
Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, MA, 1996, Chapter 12, Pages 236-257.
H. Halberstam and H. E. Richert, 1974, "Sieve methods", Academic press, London, New York, San Francisco.
|
|
|
LINKS
|
H. J. Smith, Table of n, a(n) for n = 1..20000
M. Herkommer, Goldbach Conjecture Research
Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
G. Xiao, WIMS server, Goldbach
Index entries for sequences related to Goldbach conjecture
|
|
|
FORMULA
|
From Halberstam and Richert : a(n)<(8+0(1))*c(n)*n/ln(n)^2 where c(n)=prod(p>2,(1-1/(p-1)^2))*prod(p|n,p>2,(p-1)/(p-2)). It is conjectured that the factor 8 can be replaced by 2. - Benoit Cloitre, May 16 2002
a(n) = ceil(A035026(n) / 2) = (A035026(n) + A010051(n)) / 2.
a(n) = sum_{i = 2..n} floor( 2/Omega( (i) * (2*n - i) ) ). - Wesley Ivan Hurt, Jan 24 2013
|
|
|
MATHEMATICA
|
f[n_] := Length[Select[2n - Prime[Range[PrimePi[n]]], PrimeQ]]; Table[ f[n], {n, 100}] (* Paul Abbott, Jan 11 2005 *)
nn = 10^2; ps = Boole[PrimeQ[Range[1, 2*nn, 2]]]; Join[{0, 1}, Table[Sum[ps[[i]] ps[[n-i+1]], {i, Ceiling[n/2]}], {n, 3, nn}]] (* T. D. Noe, Apr 13 2011 *)
|
|
|
PROG
|
(PARI) a(n)=my(s); forprime(p=2, n, s+=isprime(2*n-p)); s \\ Charles R Greathouse IV, Mar 27, 2012
|
|
|
CROSSREFS
|
A002375 (which differs only at the n=2 term) is the main entry for this sequence.
A023036 is the first appearance of n and A000954 is the last (assumed) appearance of n.
Cf. A185297, A187129.
Sequence in context: A094570 A225638 A002375 * A029379 A058776 A029228
Adjacent sequences: A045914 A045915 A045916 * A045918 A045919 A045920
|
|
|
KEYWORD
|
easy,nice,nonn
|
|
|
AUTHOR
|
Felice Russo
|
|
|
STATUS
|
approved
|
| |
|
|
