A272861 Sum of two integers when equal to the product of their prime-counting functions.

12, 16, 18, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 116, 120, 280, 310, 325, 330, 942, 948, 954, 960, 966, 972, 984, 990, 996, 2968, 3003, 8224, 8232, 8240, 8248, 8280, 8288, 8304, 8312, 8360, 8408, 23499, 23508, 23589

Offset

1,1

Comments

The sums are listed in increasing order. The only term with equal addends is a(2)= 16 = 8 + 8 = pi(8)^2, Indeed j=8 is the only solution to pi(j)^2 = 2*j, which is easily seen using pi(j) > j/log(j) for j>16.

Links

Giovanni Resta, Table of n, a(n) for n = 1..320 (terms < 4*10^9)

Eric Weisstein's World of Mathematics, Prime Counting Function

Formula

Positive integers n such that n = pi(j) * pi(n-j) for some j.

Example

32 is a term because 32 = 10 + 22 = 4 * 8 = pi(10) * pi(22).

Mathematica

nn = 10^3; Select[Range@ nn, Function[k, IntegerQ@ SelectFirst[Range@ nn, k == PrimePi[#] PrimePi[k - #] &]]] (* Version 10, or *)
Select[Range[10^3], Function[n, Length@ Select[Transpose@ {#, n - #} &@ Range[Floor[n/2]], n == PrimePi[First@ #] PrimePi[Last@ #] &] > 0]] (* Michael De Vlieger, Jun 30 2016 *)

Prog

(Sage) def g(n): return [k for k in (3..n/2) if n==prime_pi(k)*prime_pi(n-k)]
[n for n in range(2, 1000) if len(g(n))>0]

Crossrefs

Cf. A272860, A273286, A000720.

Sequence in context: A126763 A058080 A119911 * A192690 A064499 A080258

Adjacent sequences: A272858 A272859 A272860 * A272862 A272863 A272864

Keyword

nonn

Author

Giuseppe Coppoletta, Jun 18 2016

Extensions

a(50)-a(53) from Giovanni Resta, Jun 29 2016

Status

approved