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Sequence is undefined for n=1,2 since no composites exist whose prime divisors sum to 2, 3. For n >= 3, a(n) = A288814(prime(n)) = prime(n-k)*B(prime(n) - prime(n-k)) where B=A056240, and k >= 1 is the "type" of prime(n), indicated as prime(n)~k(g1,g2,...,gk) where gi = prime(n-(i-1)) - prime(n-i); 1 <= i <= k. Thus: 5~1(2), 211~2(12,2), 4327~3(30,8,6) etc. The sequence relates to gaps between odd primes, and in particular to the sequence of k prime gaps below prime(n). The even-indexed terms of B are relevant, as are those of subsequences:

C=A288313, 2,4 plus terms B(n) where n-3 is prime (A298252),

D=A297150, terms B(n) where n-5 is prime and n-3 is composite (A297925) and

E=A298615, terms B(n) where both n-3 and n-5 are composite (A298366).

The above sequences of indices 2m form a partition of the even numbers and the corresponding terms B(2m) form a partition of the even-indexed terms of A056240. The union of D and E is the sequence A292081 = B-C.

Let g(n,t) = prime(n) - prime(n-t), t < n, and h(n,t) = g(n,t) - g(n,1), 1 < t < n. If g1=g(n,1) is a term in A298252 (g1-3 is prime), then B(g1) is a term in C, so k=1. If g1 belongs to A297925 or A298366 then B(g1) is a term in D or E and the value of k depends on subsequent gaps below prime(n), within a range dependent on g1.

Let range R1(g1) = u - g(n,1) where u is the index in B of the greatest term in C such that C(u) < B(g1). Let range R2(g1) = v-g(n,1) where v is the index in B of the greatest term in D such that D(v) <= B(g1). For all n, R2 < R1, and if g1 is a term in D then R2(g1)=0. Examples: R1(12)=2, R2(12)=0, R1(30)=26, R2(30)=6.

k >= 1 is the smallest integer such that B(g(n,k)) <= B(g(n,t)) for all t satisfying g1 <= g(n,t) <= g1 + R1(g1). For g1-3 prime, k=1. If g1-3 is composite, let z be least integer > 1 such that g(n,z)-3 is prime, and let w be least integer >= 1 such that g(n,w)-5 is prime. Then z "complies" if h(n,z) <= R1, and w "complies" if h(n,w) <= R2. If g1-5 is prime then R2=w=0 and only z is relevant.

B(g1) must belong to C,D or E. If in C (g1-3 is prime) then k=1. If in D (g1-5 is prime), k=z if z complies, otherwise k=1. If B(g1) is in E and z complies but not w then k=z, or if w complies but not z then k=w. If B(g1) is in E and z,w both comply then k=z if 3*(g(n,z)-3) < 5*(g(n,w)-5), otherwise k=w. If neither z nor w comply, then k=1.

Conjecture: For all n >= 3, a(n) >= A288189(n).

Subsequence of A005843, same as A113935 with first term (5) excluded, since it is odd, not even. Index in A056240 of terms in A288313 (except for first two terms 2,4 of latter).

The terms in this sequence, combined with those in A297925 and A298366 form a partition of A005843(n); n>=3 (nonnegative numbers>=6). This is because any even integer n>=6 satisfies either(i) n-3 is prime, (ii) n-5 prime but n-3 composite, or (iii) n-5 and n-3 both composite.

a(n) is the smallest even number e > prime(n+1) such that e has a Goldbach partition containing prime(n+1). - Felix Fröhlich, Aug 18 2019

The sequence displays runs of consecutive even integers, whose frequency and length are related to gaps between successive primes local to these numbers. Where primes are rare (large gaps), the runs of consecutive even integers are longer (run length proportional to gap size). Let p < q be consecutive primes such that g = q-p >= 6. A string of r consecutive terms differing by 2 will start at p+7, and continue to q+1, where r = (g-4)/2. Thus at prime gap 8 a string of 2 consecutive terms differing by 2 will occur, at gap 10 there will be 3, and at gap 30 there will be 13; and so on. As the gap size increases by 2 so the run length of consecutive even terms increases by 1. The first occurrence of run length m occurs at the term corresponding to 7 + A000230(m/2).

The terms in this sequence, combined with those in A297925 and A298252 form a partition of A005843(n); n >= 3; (nonnegative even numbers >= 6). This is because any even integer n >= 6 satisfies either: (i). n-3 is prime, (ii). n-5 is prime and n-3 is composite, or (iii). both n-5 and n-3 are composite.

For any n >= 1, A056240(a(n)) = A298615(n).

This is also an ascending subsequence of the even-indexed terms of A056240(2n) (of which A292081 is a subsequence). For n >= 1, a(n) is a semiprime of the form a(n)=5*A049591(n), and the index m in A056240 of any term in this sequence belongs to the sequence of even numbers m such that m-5 is prime and m-3 is not prime (A297925). See A297925 for explanation.

For even number n, if n-5 and n-3 are both composite then A056240(n) belongs to this sequence. The union of terms in this sequence together with those in A288313 and A297150 combine to make A056240(2n), for n >= 3. A288313(n) = A056240(A298252(n)), A297150(n) = A056240(A297925(n)), and the terms of this sequence correspond to A056240(A298366). Distinct sequences A298252, A297925 and A298366 form a partition of the nonnegative even integers (A005843) >= 6. These partitions holds because any even integer n >= 6 is such that, either n-3 is prime (A298252), or n-5 is prime but n-3 is composite (A297925), or both n-5 and n-3 are composite (A298366).