

A045917


From Goldbach problem: number of decompositions of 2n into unordered sums of two primes.


61



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
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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 236257.
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,(11/(p1)^2))*prod(pn,p>2,(p1)/(p2)). 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*ni) ) ).  Wesley Ivan Hurt, Jan 24 2013


MAPLE

A045917 := proc(n)
local a, i ;
a := 0 ;
for i from 1 to n do
if isprime(i) and isprime(2*ni) then
a := a+1 ;
end if;
end do:
a ;
end proc: # R. J. Mathar, Jul 01 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[[ni+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*np)); s \\ Charles R Greathouse IV, Mar 27, 2012
(Haskell)
a045917 n = sum $ map (a010051 . (2 * n )) $ takeWhile (<= n) a000040_list
 Reinhard Zumkeller, Sep 02 2013


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.
Cf. A000040, A010051.
Sequence in context: A225638 A230443 A002375 * A240708 A235645 A240874
Adjacent sequences: A045914 A045915 A045916 * A045918 A045919 A045920


KEYWORD

easy,nice,nonn


AUTHOR

Felice Russo


STATUS

approved



