

A155216


Number of decompositions of positive even numbers 2n into unordered sums of a prime and a prime or semiprime (Chen's partitions).


3



0, 1, 2, 2, 2, 3, 3, 4, 3, 4, 5, 5, 6, 7, 4, 6, 6, 7, 8, 8, 7, 8, 9, 8, 8, 10, 9, 10, 10, 10, 13, 11, 10, 12, 11, 12, 12, 14, 12, 13, 14, 13, 13, 15, 13, 15, 15, 17, 16, 15, 15, 15, 16, 18, 16, 16, 18, 17, 19, 17, 20, 19, 19, 18, 18, 20, 19, 20, 21, 20, 18, 22, 21, 22, 20, 23, 19, 22
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OFFSET

1,3


COMMENTS

According to Chen's result, the terms of this sequence are positive, at least for sufficiently large n.


REFERENCES

J. R. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Kexue Tongbao, 17(1966), 385386.
J. R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica, 16(1973), 157176.
P. M. Ross, On Chen's theorem that each large even number has the form (p1+p2) or (p1+p2p3), J. London Math. Soc. (2) 10(1975), 500506.


LINKS



FORMULA

For n >= 2, a(n) = Sum_{3<=p<=n, p prime} A(2*n  p) + Sum_{t<=2*n, t odd semiprime} A(2*n  t) + A(n)  binomial(A(n),2) + delta(n)  a(n1)  ...  a(1), where A(n) = A033270(n), delta(n) = 1, if n is prime, and delta(n) = 2, if n is a composite number.  Vladimir Shevelev, Jul 11 2013


MAPLE

A155216 := proc(n) local a, p, q, twon ; twon := 2*n ; a := 0 ; for i from 1 do p := ithprime(i) ; if ithprime(i) > twon then break; end if; q := twon ithprime(i) ; if isprime(q) and q>= p then a := a+1 ; end if; end do: for i from 1 do p := ithprime(i) ; if ithprime(i) > twon then break; end if; q := twon ithprime(i) ; if isA001358(q) then a := a+1 ; end if; end do: return a; end proc: seq(A155216(n), n=1..80) ; # R. J. Mathar, Jul 26 2010


MATHEMATICA

a[n_] := Module[{k = 0, p, q}, For[i = 1, True, i++, p = Prime[i]; If[p > 2n, Break[]]; q = 2n  Prime[i]; If[PrimeQ[q] && q >= p, k++]]; For[i = 1, True, i++, p = Prime[i]; If[p > 2n, Break[]]; q = 2n  Prime[i]; If[ PrimeOmega[q] == 2, k++]]; k];


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



