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 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 (list; graph; refs; listen; history; text; internal format)
 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), 385-386. 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), 157-176. 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), 500-506. LINKS Peter J. C. Moses, Table of n, a(n) for n = 1..10000 V. Shevelev, Binary additive problems: recursions for numbers of representations 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(n-1) - ... - 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]; Array[a, 80] (* Jean-François Alcover, Nov 28 2017, after R. J. Mathar *) CROSSREFS Cf. A002375, A046927. Sequence in context: A070868 A342219 A272612 * A064144 A338295 A271519 Adjacent sequences: A155213 A155214 A155215 * A155217 A155218 A155219 KEYWORD nonn AUTHOR Vladimir Shevelev, Jan 22 2009 EXTENSIONS Terms beyond a(21) from R. J. Mathar, Jul 26 2010 STATUS approved

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Last modified May 30 10:07 EDT 2023. Contains 363050 sequences. (Running on oeis4.)