A102605 -id:A102605 - cp's OEIS frontend

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

Hugo Pfoertner, Oct 27 2017

nonn

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.

			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.

For references and links see A007963.

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.

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

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

A294357 Smallest odd number that can be expressed in more ways by sums of 3 odd primes p+q+r with p <= q <= r than any smaller odd number.

Original entry on oeis.org

9, 13, 17, 21, 25, 27, 29, 33, 37, 39, 45, 47, 51, 53, 63, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 161, 167, 173, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 247, 251, 257, 269, 277, 281, 287, 293, 299

Offset: 1

Hugo Pfoertner, Oct 29 2017

nonn

Position of n-th record in A007963 converted to actual odd number for which the record is achieved.
The corresponding records of numbers of representations are provided in A294358.
Empirically mod(a(n),6) = 5 for all a(n) > 63 and mod(a(n),30) != 5 for all a(n) > 425.

Hugo Pfoertner, Table of n, a(n) for n = 1..1113

Cf. A007963, A102605, A139321, A294294, A294295, A294358.

a(1)=9 because 9 = 3+3+3 is the smallest number that can be represented as sum of 3 odd primes.

a(13)=51 because A007963(25) = A007963((51-1)/2) = 14 is the 13th record in A007963.

A174847 Number m of ways of representing 2n+1 as a sum of three primes such that all 3m primes are distinct.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 3, 3, 4, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 4, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 7, 7, 6, 7, 8, 7, 7, 7, 7, 8, 8, 7, 8, 8, 8, 9, 8, 8, 9, 8, 8, 8, 8, 9, 9, 8, 9, 9, 9, 8, 9, 9, 9, 9, 10

Offset: 0

Zak Seidov, Dec 01 2010

nonn

a(n) <= A102605(n) (Number of ways of writing 2n+1 as p+q+r where p,q,r are distinct primes).
Minimal numbers with n representation as sum of triple of primes such that all 3n primes are distinct are:
15,29,49,71,91,119,137,167,189,227,
255,273,317,345,375,369,435,483,495,535,
567,597,641,651,699,731,755,791,821,867,921,975.

			First number with m=1 is 15=3+5+7; for m=2,3,4 we have:
m=2: 29=3+7+19=5+11+13;  m=3: 49=3+5+41=5+7+37=13+17+19; m=4: 71=3+7+61=5+13+53=7+11+53=13+17+41.

Cf. A102605.

cp's OEIS Frontend

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).

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A294357 Smallest odd number that can be expressed in more ways by sums of 3 odd primes p+q+r with p <= q <= r than any smaller odd number.

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A174847 Number m of ways of representing 2n+1 as a sum of three primes such that all 3m primes are distinct.

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