

A322007


a(n) = number of primes of the form p = 2n  q, where q is a prime or semiprime.


2



0, 0, 1, 2, 3, 3, 4, 4, 6, 5, 6, 7, 8, 8, 9, 7, 8, 9, 11, 9, 11, 11, 11, 12, 13, 12, 13, 14, 13, 13, 16, 15, 16, 16, 14, 16, 18, 16, 19, 19, 17, 18, 21, 17, 19, 22, 19, 19, 24, 19, 21, 23, 20, 21, 26, 22, 23, 28, 23, 24, 29, 23, 24, 29, 21, 25, 29, 24, 25, 29, 27, 25, 33, 26, 27, 32, 27
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OFFSET

0,4


COMMENTS

Related to Chen's theorem (Chen 1966, 1973) which states that every sufficiently large even number is the sum of a prime and another prime or semiprime. Yamada (2015) has proved that this holds for all even numbers larger than exp(exp(36)).
In terms of this sequence, Chen's theorem with Yamada's bound is equivalent to say that a(n) > 0 for all n > 1.7 * 10^1872344071119348 (exponent ~ 1.8*10^15).
A235645 lists the number of decompositions of 2n into a prime p and a prime or semiprime q; this is less than a(n) because p + q and q + p is the same decomposition (if q is a prime), but this sequence will count two distinct primes 2n  q and 2n  p (if q <> p).
Sequence A322006 lists the same for even and odd numbers n, not only for even numbers 2n.


REFERENCES

Chen, J. R. (1966). "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao. 11 (9): 385386.
Chen, J. R. (1973). "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: 157176.


LINKS



FORMULA



EXAMPLE

a(4) = 2 since for n = 4, 2n = 8 = 2 + 6 = 3 + 5 = 5 + 3, i.e., primes 2, 3 and 5 are of the form specified in the definition (since 6 = 2*3 is a semiprime and 5 and 3 are primes).


PROG

(PARI) A322007(n, s=0)=forprime(p=2, 2+n*=2, bigomega(np)<3&&s++); s}


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



