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Conjecture: a(n)>0 for all n>8. Moreover, for positive integers a<=b<=c, all integers n>=3(a+b+c) with n-a-b-c even can be written as a*p+b*q+c*r with p,q,r terms of A210479, if and only if (a,b,c) is among the following 6 triples: (1,2,3), (1,2,4), (1,2,8), (1,2,9), (1,3,5), (1,3,8).

The author also conjectured that if n>8 is odd, different from 201 and 447, and not congruent to 1 or -1 modulo 12, then n can be written as a sum of three terms of A210479.

Melfi proved that this sequence is infinite.

Conjecture: a(n) exists for any n > 0. Moreover, any positive rational number r can be written as q'/q, where q and q' are terms of A258838 (i.e., q is practical with q-1 and q+1 twin prime, and q' is practical with q'-1 and q'+1 twin prime).

This implies that there are infinitely many "sandwiches of the second kind" (i.e., triples {q-1,q,q+1} with q practical and q-1 and q+1 twin prime).

I have verified the conjecture for all those rational numbers r = n/m with m,n = 1,...,1000. -Zhi-Wei Sun, Jun 15 2015

The author introduced two kinds of "sandwiches" in 2013. The conjecture in A258836 essentially says that {a(m)/a(n): m,n = 1,2,3,...} coincides with the set of all positive rational numbers. This implies that the sequence contains infinitely many terms.

Conjecture: a(n)>0 for all n>3.

This is stronger than Goldbach's conjecture and the author's conjecture that any odd number greater than one is the sum of a prime and a practical number. Also, it implies that there are infinitely many primes p with p-1 and p+1 both practical.

The author has verified this new conjecture for n up to 10^7.

Conjecture: a(n) > 0 for all n > 4.

This conjecture involves two kinds of sandwiches introduced by the author, and it is much stronger than the Goldbach conjecture for odd numbers. We have verified the conjecture for n up to 10^7.

Zhi-Wei Sun also made the following conjectures:

(1) Any even number greater than 10 can be written as the sum of four elements in the set

S = {prime p: p-1 and p+1 are both practical}.

Also, every n=3,4,5,... can be represented as the sum of a prime in S and two triangular numbers.

(2) Each integer n>7 can be written as p + q + x^2 (or p + q + x(x+1)/2), where p is a prime with p-1 and p+1 both practical, and q is a practical number with q-1 and q+1 both prime.

(3) Every n=3,4,... can be written as the sum of three elements in the set

T = {x: 6x is practical with 6x-1 and 6x+1 both prime}.

(4) Any integer n>6 can be represented as the sum of two elements of the set S and one element of the set T.

(5) Each odd number greater than 11 can be written in the form 2p+q+r, where p and q belong to S, and r is a practical number with r-1 and r+1 both prime.

Conjecture: a(n)<=3n for all n>0. Moreover, a(2n-1)/(2n-1) and a(2n)/(2n) have limits 1 and 2 respectively, as n tends to the infinity.

Conjecture: a(n)>0 for all n>5.

This has been verified for n up to 300000.

Note that for n=406 we cannot represent n in the given way with q+1 practical.

Conjecture: a(n)>0 for all n>2.

This has been verified for n up to 10^7.

Let s_n=sum_{k=1}^n(-1)^{n-k}p_k for n=1,2,3,... The author also made the following conjectures:

(1) For each n>2, there is an integer k>0 such that 3(n-s_k)-1 and 3(n-s_k)+1 are twin primes.

(2) For each n>3, there is an integer k>0 such that 3(n-s_k)-2 and 3(n-s_k)+2 are cousin primes.

(3) Every n=6,7,... can be written as p+s_k (k>0) with p and p+6 sexy primes.

(4) Any integer n>3 different from 65 and 365 can be written as p+s_k (k>0) with p a term of A210479.

(5) Each integer n>8 can be written as q+s_k (k>0) with q-4, q, q+4 all practical.

(6) Any integer n>1 can be written as j(j+1)/2+s_k with j,k>0.

Conjecture: All the terms are positive.

See also the comments related to A222579.