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%I #15 Sep 08 2022 08:45:23
%S 0,1,2,4,5,6,11,9,8,20,14,14,26,17,18,48,22,22,49,28,36,69,33,37,68,
%T 47,43,83,49,47,125,50,53,118,56,94,126,63,63,153,98,71,186,79,94,230,
%U 89,91,197,127,127,215,112,105,220,172,147
%N Number of distinct representations of (2n)^2 as the sum of two primes.
%C 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.
%C a(n)=A061358(4n^2). - _Emeric Deutsch_, Apr 03 2006
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
%D H. Halberstam and H. E. Richert, 1974, "Sieve methods", Academic press, London, New York, San Francisco.
%H Marius A. Burtea, <a href="/A113631/b113631.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = A045917(2n). a(n) = #{p(i) + p(j) = A016742(n) for p(k) = A000040(k) and i >= j}. a(n) = #{p(i) + p(j) = (2*n)^2 for p(k) = A000040(k) and i >= j}.
%e a(1) = 1 because (2*1)^2 = 4 = 2 + 2 uniquely.
%e a(2) = 2 because (2*2)^2 = 16 = 3 + 13 = 5 + 11.
%e a(3) = 4 because (2*3)^2 = 36 = 5 + 31 = 7 + 29 = 13 + 23 = 17 + 19.
%e a(4) = 5 because (2*4)^2 = 64 = 3 + 61 = 5 + 59 = 11 + 53 = 17 + 47 = 23 + 41.
%e a(5) = 6 because (2*5)^2 = 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53.
%e 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.
%p 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
%o (Magma) [#RestrictedPartitions(4*n^2,2,{p:p in PrimesUpTo(20000)}):n in [0..56] ] // _Marius A. Burtea_, Jan 19 2019
%Y Cf. A000040, A045917, A016742.
%Y Cf. A061358.
%K easy,nonn
%O 0,3
%A _Jonathan Vos Post_, Mar 31 2006
%E Corrected and extended by _Emeric Deutsch_, Apr 03 2006
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