A129363 Number of partitions of 2n into the sum of two twin primes.

0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 1, 2, 2, 3, 3, 2, 1, 2, 2, 3, 4, 2, 1, 2, 1, 2, 3, 3, 2, 2, 1, 2, 4, 3, 3, 4, 2, 2, 3, 2, 2, 4, 2, 0, 0, 0, 2, 4, 3, 2, 2, 2, 4, 6, 3, 3, 5, 3, 1, 2, 1, 2, 4, 2, 1, 2, 2, 4, 5, 3, 2, 4, 3, 3, 4, 2, 2, 4, 2, 3, 6, 3, 1, 2, 1, 3, 6, 4, 2, 2, 1, 2, 4, 3, 4, 6, 4, 4, 5, 3, 6, 12

Offset: 1

Author

T. D. Noe, Apr 11 2007

Keywords

• nonn

Comments

a(n/2)=0 for the n in A007534. The logarithmic plot of this sequence seems very regular after 200000 terms.

Examples

			a(11)=3 because 22 = 3+19 = 5+17 = 11+11.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• James Grime and Brady Haran, Goldbach Conjecture (but with TWIN PRIMES), Numberphile video (2024)
• T. D. Noe, Logarithmic plot of 10^6 terms

Crossrefs

Cf. A175931 (n for which a(n-1), a(n), a(n+1) are equal).

Cf. A001097, A002375, A007534.

Programs

Haskell

a129363 n = sum $ map (a164292 . (2*n -)) $ takeWhile (<= n) a001097_list
-- Reinhard Zumkeller, Feb 03 2014

Mathematica

nn=1000; tw=Select[Prime[Range[PrimePi[nn]]], PrimeQ[ #+2]&]; tw=Union[tw,tw+2]; tc=Table[0,{nn}]; tc[[tw]]=1; Table[cnt=0; k=1; While[tw[[k]]<=n/2, cnt=cnt+tc[[n-tw[[k]]]]; k++ ]; cnt, {n,2,nn,2}]

Formula

a(n) = Sum_{i=1..n} ceiling((A010051(i+2) + A010051(i-2))/2) * ceiling((A010051(2n-i+2) + A010051(2n-i-2))/2) * A010051(2n-i) * A010051(i). - Wesley Ivan Hurt, Jan 30 2014

a(n) = sum(A164292(2*n - A001097(k)): A001097(k) <= n). - Reinhard Zumkeller, Feb 03 2014

Extensions

Comment converted to crossref by Klaus Brockhaus, Oct 27 2010