A155216 Number of decompositions of positive even numbers 2n into unordered sums of a prime and a prime or semiprime (Chen's partitions).

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

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