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%I #71 Mar 07 2020 14:56:51
%S 12,18,24,96,116,120,984,990,996,8408,23616,23742,23850,24030,24066,
%T 24084,480324,480336,481344,3523814,3523842,3523884,3524514,9557160,
%U 9558030,9558240,9558300,25874592,25874640,70119798,189960894,189961344,189962352,189963594,189963630,189969102
%N Sums of two primes (in increasing order) when equal to the product of their prime-counting functions.
%C 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.
%C Primes p,q can only occur for a finite number of terms n (see comments in A273286).
%C Conjecture: the sequence is infinite and each term has only one decomposition into a sum of suitable primes p,q.
%C From _David A. Corneth_, Jun 28 2016: (Start)
%C 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.
%C 3, 4, 8
%C 4, 24, 30
%C 6, 164, 166
%C 8, 1051, 1051
%C 9, 2624, 2676
%C 12, 40027, 40112
%C Can these bounds on pi(q) be expressed in terms of pi(p)? (End)
%H Giuseppe Coppoletta, <a href="/A272860/b272860.txt">Table of n, a(n) for n = 1..43</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RossersTheorem.html">Rosser's Theorem</a>
%H Pierre Dusart, <a href="http://arxiv.org/abs/1002.0442/">Estimates of some functions over primes without R.H.</a>, arXiv:1002.0442 [math.NT], 2010.
%F Numbers n = p+q = pi(p)*pi(q) for some primes p and q.
%F Equivalently, n = i*j = prime(i) + prime(j) for some i,j.
%F A272862 gives the corresponding terms pi(q) (with q>p). The terms pi(p) are given by A272860 / A272862
%e 12 is a term because 12 = 5 + 7 = pi(5) * pi(7).
%t 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 *)
%o (Sage)
%o def sol(n):
%o 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]
%o N=25000
%o v=[n for n in range(2,N,2) if len(sol(n))>0]
%o print('A272862 =',v)
%o list_pi=flatten([sol(n) for n in range(2,N,2) if sol(n)])
%o print('list_pi(p) =',list_pi)
%Y Cf. A272861, A272862, A273286, A000040, A000720.
%K nonn
%O 1,1
%A _Giuseppe Coppoletta_, Jun 19 2016
%E More terms from _David A. Corneth_, Jun 28 2016
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