

A272860


Sums of two primes (in increasing order) when equal to the product of their primecounting functions.


4



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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(nnth_prime(k)) and k*prime_pi(nnth_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 len(sol(n))>0]); 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



