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

1,1

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.

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.

Formula

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

Example

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.

Crossrefs

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

Sequence in context: A176852 A248630 A130569 * A078176 A097494 A037136

Adjacent sequences: A294291 A294292 A294293 * A294295 A294296 A294297

Keyword

nonn

Author

Hugo Pfoertner, Oct 27 2017

Status

approved