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A113631 Number of distinct representations of (2n)^2 as the sum of two primes. 3
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, 47, 43, 83, 49, 47, 125, 50, 53, 118, 56, 94, 126, 63, 63, 153, 98, 71, 186, 79, 94, 230, 89, 91, 197, 127, 127, 215, 112, 105, 220, 172, 147 (list; graph; refs; listen; history; text; internal format)
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

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}.

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

Cf. A000040, A045917, A016742.

Cf. A061358.

Sequence in context: A165701 A129305 A278507 * A202101 A192582 A101951

Adjacent sequences:  A113628 A113629 A113630 * A113632 A113633 A113634

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Mar 31 2006

EXTENSIONS

Corrected and extended by Emeric Deutsch, Apr 03 2006

STATUS

approved

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Last modified November 13 00:25 EST 2019. Contains 329083 sequences. (Running on oeis4.)