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).

7, 11, 15, 19, 23, 25, 31, 35, 37, 43, 45, 49, 55, 61, 63, 69, 75, 79, 81, 85, 87, 91, 99, 105, 111, 117, 129, 135, 141, 147, 159, 165, 171, 177, 195, 201, 207, 219, 225, 231, 237, 255, 261, 267, 279, 285, 291, 297, 309, 315, 321, 339, 345, 351

Offset: 1

Author

Hugo Pfoertner, Oct 27 2017

Keywords

• nonn

Comments

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.

The corresponding minimum numbers of representations are provided in A294295.

Empirically, mod(a(n),6) = 3 for all a(n) > 91 and mod(a(n),30) = 15 for all a(n) > 1281.

Examples

			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).
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.
		

References

• For references and links see A007963.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..301
• H. A. Helfgott, The ternary Goldbach conjecture is true, arXiv:1312.7748 [math.NT], 2013-2014.

Crossrefs

Cf. A007963, A102605, A139321, A294295, A294357, A294358.

Formula

A007963(k) > A007963((a(n)-1)/2) for all k > (a(n)-1)/2.