A072511 - OEIS

A072511

Least number m such that 2m can be expressed as the sum of two distinct primes in exactly n ways.

1, 4, 8, 12, 18, 24, 30, 39, 42, 45, 57, 72, 60, 84, 90, 117, 123, 144, 120, 105, 162, 150, 180, 237, 165, 264, 288, 195, 231, 240, 210, 285, 255, 336, 396, 378, 438, 357, 399, 345, 519, 315, 504, 465, 390, 480, 435, 462, 450, 567, 717, 420, 495, 651, 540, 615

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OFFSET

0,2

COMMENTS

Let f(x) = A061357(x) be the number of primes p < x such that 2x-p is also prime. a(n) is the smallest positive integer x such that f(x) = n.

Or, least number m such that m can be expressed as the mean of two distinct primes in exactly n ways. Cf. A061357 = number of ways n can be expressed as the mean of two distinct primes, A061357 = number of ways the even integer 2n can be written as the sum of two primes for all even integers >6. - Zak Seidov, Sep 08 2006

For what values of n is a(n) > a(n+1)?

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

FORMULA

It seems that for n>7 n*log(n)*log(log(n)) < a(n) < 3n*log(n)*log(log(n)). Does lim n->infinity a(n)/n/log(n)/log(log(n)) exist ? - Benoit Cloitre, Aug 11 2002

EXAMPLE

a(1)=4 because 8 = 3+5 that is 8 can be expressed as the sum of two distinct primes by exactly 1 way,

a(2)=8 because 16 = 3+13 = 5+11 (2 ways),

a(3)=12 because 24 = 5+17 = 7+17 = 11+17 (3 ways),

a(4)=18 because 36 = 5+31 = 7+29 = 13+23 = 17+19 (4 ways), etc.

Starting with third term 12, all terms are multiples of 3.

MATHEMATICA

f[x_] := Length[Select[2x-(Prime/@Range[PrimePi[x-1]]), PrimeQ]]; For[x=1, x<1000, x++, fx=f[x]; If[a[fx]>=0, Null, Null, a[fx]=x]]; a/@Range[0, 60]

PROG

(Haskell)

import Data.List (elemIndex)

import Data.Maybe (fromJust)