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A294294 Conjecturally, all odd numbers greater than a(n) can be represented in more ways by the sum of 3 odd primes p+q+r with p<=q<=r than a(n).

%I A294294 #19 May 03 2019 08:42:19
%S A294294 7,11,15,19,23,25,31,35,37,43,45,49,55,61,63,69,75,79,81,85,87,91,99,
%T A294294 105,111,117,129,135,141,147,159,165,171,177,195,201,207,219,225,231,
%U A294294 237,255,261,267,279,285,291,297,309,315,321,339,345,351
%N A294294 Conjecturally, all odd numbers greater than a(n) can be represented in more ways by the sum of 3 odd primes p+q+r with p<=q<=r than a(n).
%C A294294 The sequence provides numerical evidence of the validity of the ternary Goldbach conjecture, i.e. that every odd number >5 can be written as the sum of 3 primes, now proved by A. Helfgott.
%C A294294 The corresponding minimum numbers of representations are provided in A294295.
%C A294294 Empirically, mod(a(n),6) = 3 for all a(n) > 91 and mod(a(n),30) = 15 for all a(n) > 1281.
%D A294294 For references and links see A007963.
%H A294294 Hugo Pfoertner, <a href="/A294294/b294294.txt">Table of n, a(n) for n = 1..301</a>
%H A294294 H. A. Helfgott, <a href="https://arxiv.org/abs/1312.7748">The ternary Goldbach conjecture is true</a>, arXiv:1312.7748 [math.NT], 2013-2014.
%F A294294 A007963(k) > A007963((a(n)-1)/2) for all k > (a(n)-1)/2.
%e A294294 a(1)=7 because all odd numbers > 7 have more representations by sums of 3 odd primes than 7, which has no such representation (A294295(1)=0).
%e A294294 a(2)=11, because all odd numbers > 11 have at least 2 representations p+q+r, e.g. 13=3+3+7=5+5+3 whereas 11=3+3+5 and 9=3+3+3 only have A294295(2)=1 representation.
%Y A294294 Cf. A007963, A102605, A139321, A294295, A294357, A294358.
%K A294294 nonn
%O A294294 1,1
%A A294294 _Hugo Pfoertner_, Oct 27 2017