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A272861 Sum of two integers when equal to the product of their prime-counting functions. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 17 02:56 EDT 2019. Contains 325092 sequences. (Running on oeis4.)