A272860 Sums of two primes (in increasing order) when equal to the product of their prime-counting functions.

12, 18, 24, 96, 116, 120, 984, 990, 996, 8408, 23616, 23742, 23850, 24030, 24066, 24084, 480324, 480336, 481344, 3523814, 3523842, 3523884, 3524514, 9557160, 9558030, 9558240, 9558300, 25874592, 25874640, 70119798, 189960894, 189961344, 189962352, 189963594, 189963630, 189969102

Offset: 1

Author

Giuseppe Coppoletta, Jun 19 2016

Keywords

• nonn

Comments

Each term is necessarily even and 3 < p < q in the formula n = p+q = pi(p)*pi(q). Indeed, assuming p<=q, if p=2 then n = 2+q = pi(2)*pi(q) = pi(q) < q. Inequality p > 3 easily follows from prime(k) > k*log(k) and if p=q then 2*p = pi(p)^2 with no solution.

Primes p,q can only occur for a finite number of terms n (see comments in A273286).

Conjecture: the sequence is infinite and each term has only one decomposition into a sum of suitable primes p,q.

From David A. Corneth, Jun 28 2016: (Start)

Pi(p) and pi(q) seem dependent on each other. Below is a small list of pi(p), the least corresponding pi(q) and the largest corresponding pi(q). If a value of pi(p) isn't listed, no terms are formed with it.

3, 4, 8

4, 24, 30

6, 164, 166

8, 1051, 1051

9, 2624, 2676

12, 40027, 40112

Can these bounds on pi(q) be expressed in terms of pi(p)? (End)

Examples

			12 is a term because 12 = 5 + 7 = pi(5) * pi(7).
		

Links

• Giuseppe Coppoletta, Table of n, a(n) for n = 1..43
• Eric Weisstein's World of Mathematics, Rosser's Theorem
• Pierre Dusart, Estimates of some functions over primes without R.H., arXiv:1002.0442 [math.NT], 2010.

Crossrefs

Cf. A272861, A272862, A273286, A000040, A000720.

Programs

Mathematica

Select[Range[10^3], Function[n, MemberQ[Times @@ # & /@ PrimePi@ Select[Transpose@ {#, n - #} &@ Range[Floor[n/2]], Times @@ Boole@ PrimeQ@ {First@ #, Last@ #} == 1 &], n]]] (* Michael De Vlieger, Jun 29 2016 *)

Sage

def sol(n):
    return [k for k in divisors(n) if k^2<= n and is_prime(n-nth_prime(k)) and k*prime_pi(n-nth_prime(k))==n]
N=25000
v=[n for n in range(2,N,2) if len(sol(n))>0]
print('A272862 =',v)
list_pi=flatten([sol(n) for n in range(2,N,2) if sol(n)])
print('list_pi(p) =',list_pi)

Formula

Numbers n = p+q = pi(p)*pi(q) for some primes p and q.

Equivalently, n = i*j = prime(i) + prime(j) for some i,j.

A272862 gives the corresponding terms pi(q) (with q>p). The terms pi(p) are given by A272860 / A272862

Extensions

More terms from David A. Corneth, Jun 28 2016