OFFSET
0,3
COMMENTS
From Halberstam and Richert: A045917(2n)<(8+0(1))*c(n)*n/log(n)^2 where c(n)=prod(p>2,(1-1/(p-1)^2))*prod(p|n,p>2,(p-1)/(p-2)). Hence a(n) = A045917(2n) < (8+0(1))*c(2n)*2n/log(2n)^2 where c(k)=prod(p>2,(1-1/(p-1)^2))*prod(p|k,p>2,(p-1)/(p-2)). See also: A045917 From Goldbach problem: number of decompositions of 2n into unordered sums of two primes. A016742 Even squares: (2n)^2.
a(n)=A061358(4n^2). - Emeric Deutsch, Apr 03 2006
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
H. Halberstam and H. E. Richert, 1974, "Sieve methods", Academic press, London, New York, San Francisco.
LINKS
Marius A. Burtea, Table of n, a(n) for n = 0..500
FORMULA
EXAMPLE
a(1) = 1 because (2*1)^2 = 4 = 2 + 2 uniquely.
a(2) = 2 because (2*2)^2 = 16 = 3 + 13 = 5 + 11.
a(3) = 4 because (2*3)^2 = 36 = 5 + 31 = 7 + 29 = 13 + 23 = 17 + 19.
a(4) = 5 because (2*4)^2 = 64 = 3 + 61 = 5 + 59 = 11 + 53 = 17 + 47 = 23 + 41.
a(5) = 6 because (2*5)^2 = 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53.
a(6) = 11 because (2*6)^2 = 144 = 5 + 139 = 7 + 137 = 13 + 131 = 17 + 127 = 31 + 113 = 37 + 107 = 41 + 103 = 43 + 101 = 47 + 97 = 61 + 83 = 71 + 73.
MAPLE
g:=sum(sum(x^(ithprime(i)+ithprime(j)), i=1..j), j=1..1500): gser:=series(g, x=0, 12560): 0, seq(coeff(gser, x^(4*n^2)), n=1..56); # Emeric Deutsch, Apr 03 2006
PROG
(Magma) [#RestrictedPartitions(4*n^2, 2, {p:p in PrimesUpTo(20000)}):n in [0..56] ] // Marius A. Burtea, Jan 19 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Mar 31 2006
EXTENSIONS
Corrected and extended by Emeric Deutsch, Apr 03 2006
STATUS
approved