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 A269329 Number of partitions of a positive integer n into two distinct primes such that for even n, it is of the form n = p + q and for odd n, it is of the form n = 2p' + q'. 0
 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 4, 2, 2, 2, 2, 2, 3, 3, 3, 2, 4, 2, 4, 3, 2, 2, 4, 3, 3, 4, 4, 1, 3, 3, 5, 4, 4, 3, 6, 3, 4, 5, 4, 4, 6, 3, 5, 5, 3, 3, 6, 3, 3, 6, 3, 2, 7, 5, 7, 6, 5, 2, 6, 5, 4, 6, 4, 4, 8, 5, 7, 7, 5, 4, 8, 4, 4, 8, 7, 4, 7, 4, 7, 9, 4, 4, 10, 4, 5, 7, 6, 3, 9 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,11 COMMENTS This sequence combines Levy's conjecture for odd positive numbers with the Goldbach conjecture for even positive numbers and strenghtens both by restricting the prime pairs to be distinct. I.e., every positive integer n > 6 is the sum of two distinct primes p and q such that for n even, it is of the form n = p + q and for n odd, it is of the form n = 2p' + q'. LINKS EXAMPLE a(23)=3. Hence there are 3 partitions (as defined above) of the odd integer 23, namely 19+2+2, 17+3+3 and 13+5+5. a(24)=3. Hence there are 3 partitions of the even integer 24, namely 19+5, 17+7 and 13+11. MATHEMATICA parts[n_, a_, b_] := Select[IntegerPartitions[n, {a+b}, Prime@Range[PrimePi[n]]], Length[Union@#]==2&&MemberQ[Values@Counts@#, a] &]; lst1=Table[Length@parts[2n-1, 1, 2], {n, 1, 200}]; lst2=Table[Length@parts[2n, 1, 1], {n, 1, 200}]; Riffle[lst1, lst2] CROSSREFS Cf. A002375, A061357, A194830, A223893, A225522. Sequence in context: A280444 A030422 A090001 * A117166 A078996 A084610 Adjacent sequences:  A269326 A269327 A269328 * A269330 A269331 A269332 KEYWORD nonn AUTHOR Frank M Jackson and M. B. Rees, Feb 23 2016 STATUS approved

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Last modified June 16 11:12 EDT 2019. Contains 324152 sequences. (Running on oeis4.)