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

1,1

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)

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.

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

Example

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

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 *)

Prog

(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)

Crossrefs

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

Sequence in context: A224218 A076485 A325387 * A071354 A006622 A124269

Adjacent sequences: A272857 A272858 A272859 * A272861 A272862 A272863

Keyword

nonn

Author

Giuseppe Coppoletta, Jun 19 2016

Extensions

More terms from David A. Corneth, Jun 28 2016

Status

approved