cp's OEIS Frontend

Conjecture: a(n)>0 for all n>0. Moreover, for any positive integers m and n, if m is greater than one or n is not among 17, 181, 211, 251, 313, 337, then 2n can be written as p+q with p, q and p^{3m}+q^{3m} all practical.

This conjecture implies that for each m=1,2,3,... there are infinitely many practical numbers of the form p^{3m}+q^{3m} with p and q both practical.

Conjecture: a(n)>0 for all n>0. Moreover, if n>0 is different from 74, 138, 166, 542, then n+k^3 is practical for some 0<=k<=sqrt(n)*log(n); if n is not equal to 102, then n+k and n+k^3 are both practical for some k=0,...,n-1.

Zhi-Wei Sun also conjectured that any integer n>1 can be written as x^3+y (x,y>0) with 2x and 4xy both practical.

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>2. Moreover, for each m=2,3,4,... any sufficiently large even integer can be written as x+y (x,y>0) with x-1 and x+1 both prime, and x and x^m+y^m both practical.

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)>0 except for n = 1,...,8, 10, 520, 689, 740.

Zhi-Wei Sun also guessed that any integer n>6 different from 407 can be written as p+q+F_k, where p is a prime with p-1 and p+1 practical, q is a practical number with q-1 and q+1 prime, and F_k (k>=0) is a Fibonacci number.

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 > 6. Also, for any integer n > 2, there is a prime p < n such that n-(p-1) and n+(p-1) are both prime or both practical.

Note that 1 is the only odd practical number and 2 is the only even prime.