The author has verified this for n up to 10^8, and also guessed the following refinement: If n>6 is not among 20, 104, 272, 464, 1664, then n can be written as p+q with p an even practical number and q a positive triangular number.
Somu and Tran (2024) proved the conjecture that a(n)>0 for n>0. - Duc Van Khanh Tran, Apr 24 2024
The author has verified this for n up to 2*10^8. It is known that there are infinitely many practical numbers q with q+2 also practical.
Zhi-Wei Sun also made the following similar conjectures:
(1) Each odd number n>5 can be written as p+q with p and p+6 both prime and q practical. Also, any odd number n>3 not equal to 55 can be written as p+q with p and p+2 both prime and q practical.
(2) Each integer n>10 can be written as x+y (x,y>0) with 6x-1 and 6x+1 both prime, and y and y+6 both practical.
Also, any integer n>=6360 can be written as x+y (x,y>0) with 6x-1 and 6x+1 both prime, and y and y+2 both practical.
a(n) = [A030057(n) > n], where [ ] is the Iverson bracket.
PROG
(PARI) A322860 = is_A005153 \\ Please do not duplicate code by pasting copies from one place to others, it will be near to impossible to track down all copies if something has to be corrected or improved. - M. F. Hasler, Jun 19 2023
Different from A007621: A007621 contains no odd numbers, while every odd term in A083207 is here. The numbers 738, 748, 774, 846, ... are in A007621 and are not here.
But the subsequence of even terms (A005843 intersect this sequence) is a subsequence of A007621:
70 is a term since 70 is a Zumkeller number but not a practical number: 1+5+7+10+14+35 = 2+70, so 70 is a Zumkeller number; but 4 cannot be written as a sum of distinct divisors of 70, so 70 is not practical.
Largest primitive practical number p that divides the n-th practical number - A005153(n) such that the radical of the quotient A005153(n)/p is a divisor of p.
The relationship between a practical number and the largest primitive practical number that divides it such that the radical of the quotient is a divisor of same primitive provides an equivalence relation. Practical numbers that have the same progenitive primitive. This property arises from the characteristic of practical numbers and, in particular, primitives that says a practical number multiplied by power combinations of any of its divisors is also practical. This sequence identifies the primitive progenitor of each practical number A005153(n).
Note that this sequence and A378202 are similar but the first difference is at a(63) as explained in the example.
a(63) = 66. A005153(63) = 264 and the largest primitive practical number that divides the practical number 264 is 88. However the radical of the quotient 264/88 is 3 and 3 is not a divisor of 88. The next greatest primitive divisor of 264 is 66 and the radical of the quotient 264/66 is 2 and 2 is a divisor of 66.
a(131) = 306. A005153(131) = 612 and it is divisible by two primitive practical numbers 204 and 306 with their quotient a divisor of their primitive in both cases but 306 is chosen as the larger primitive.
According to Margenstern and proved by Weingartner (see links) the density of practical numbers is greater than the density of primes. Margenstern calculated that the density of practical numbers was approx 1.2767 (1.3411/1.059) times greater than the density of primes in the interval 1 to 10^12. This sequence shows that the set of places where no practical number exists between successive primes has a degree of regularity and appears to be infinite.
A conjecture by Zhi-Wei Sun states that any rational number can be expressed as the sum of distinct unit fractions whose denominators are practical numbers. To prove this conjecture, David Eppstein (see link) used the fact that every natural number when repeatedly multiplied by 2 will eventually become practical.
Let r_m be the natural density of the set of integers n with a(n) = m. Then r_m is positive if and only if m is practical. In that case, r_m = (1/m)*P_m, where P_m is the product of (1-1/p) over primes p <= sigma(m) + 1 (see Cor. 1 of Weingartner 2015). The first few values of (m, r_m) are (1, 1/2), (2, 1/6), (4, 2/35), (6, 32/1001), (8, 24/1001), (12, 36864/2800733), ...
As y grows, the natural density of integers n, which satisfy a(n) > y, is asymptotic to c*exp(-gamma)/log(y), where c = 1.33607... is the constant factor in the asymptotic for the count of practical numbers (A005153) and gamma = 0.577215... is Euler's constant (see Eq. (3) of Weingartner (2015)). For example, about 1% of integers n satisfy a(n) > exp(75), because c*exp(-gamma)/75 = 0.010... (End)
If n = Product_{i=1..r} p_i^e_i, then define n_0 = 1, n_j = Product_{i=1..j} p_i^e_i. a(n) = n_j where j is the first index for which p_{j+1} > sigma(n_j) + 1, or j = r if no such index exists.
