A272861 -id:A272861 - cp's OEIS frontend

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

Giuseppe Coppoletta, Jun 19 2016

nonn

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)

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

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.

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

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 *)

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)

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

More terms from David A. Corneth, Jun 28 2016

A272862 Positive integers j such that prime(i) + prime(j) = i*j for some i <= j.

Original entry on oeis.org

4, 6, 8, 24, 29, 30, 164, 165, 166, 1051, 2624, 2638, 2650, 2670, 2674, 2676, 40027, 40028, 40112, 251701, 251703, 251706, 251751, 637144, 637202, 637216, 637220, 1617162, 1617165, 4124694, 10553383, 10553408, 10553464, 10553533, 10553535, 10553839, 69709686

Offset: 1

Giuseppe Coppoletta, Jul 25 2016

nonn

Also pi(q) for primes q verifying p+q = pi(p)*pi(q) for some prime p <= q.

The list of products i*j gives A272860. See also comments there.

			8 is a term as prime(3) + prime(8) = 3*8.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..43

Cf. A272860, A272861, A000720, A000040.

Select[Range[3000], Function[j, Total@ Boole@ Map[Prime@ # + Prime@ j == # j &, Range@ j] > 0]] (* Michael De Vlieger, Jul 28 2016 *)

is(n) = for(i=1, n, if(prime(i)+prime(n)==i*n, return(1))); return(0) \\ Felix Fröhlich, Jul 27 2016

is(n,p=prime(n))=my(i); forprime(q=2,p, if(i++*n==p+q, return(1))); 0
v=List(); n=0; forprime(p=2,1e6, if(is(n++,p), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jul 28 2016

def sol(n):
    if n<5: a=n
    else: a=exp(n+1)/(n+1)
    b=(n-1)/n^2*exp(n^2/(n-1.1))
    return [j for j in range(a,b) if is_prime(n*j-nth_prime(n)) and prime_pi(n*j-nth_prime(n))==j]
flatten([sol(i) for i in (1..15) if len(sol(i))>0]) #

cp's OEIS Frontend

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

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