A025019 -id:A025019 - cp's OEIS frontend

A245077 Largest k such that the smallest prime satisfying Goldbach's conjecture is less than or equal to (2n)^(1/k).

2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 3, 3, 2, 1, 3, 2, 3, 3, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 2, 4, 2, 4, 2, 2, 4, 2, 2, 1, 4, 2, 4, 4, 2, 4, 4, 2, 4, 2, 2, 1, 2, 1, 1, 4

Offset: 2

Jon Perry, Jul 11 2014

nonn

The 1's appear as in A244408.

			For n=5 we have 3+7=10. As rt3(10)<3<sqrt(10), a(5)=2.

Jens Kruse Andersen, Table of n, a(n) for n = 2..10000

Cf. A244408.

Cf. A020481, A002373, A025018, A025019.

for (n=2, 100, p=2; while(!isprime(2*n-p), p=nextprime(p+1)); k=1; while(p<=(2*n)^(1/k), k++); print1(k-1", ")) \\ Jens Kruse Andersen, Jul 12 2014

Definition corrected by Jens Kruse Andersen, Jul 12 2014

A274189 Even numbers 2n that satisfy an extended Goldbach conjecture: They have at least one Goldbach partition 2n = p + q (p <= q; p, q prime) that satisfies p <= sqrt(n), at least one with sqrt(n) < p <= sqrt(2n) and at least one with p > sqrt(2n).

Original entry on oeis.org

34, 46, 50, 66, 74, 78, 86, 138, 142, 160, 162, 168, 170, 174, 176, 178, 180, 184, 186, 194, 202, 204, 206, 234, 236, 238, 240, 242, 246, 252, 254, 264, 270, 276, 282, 284, 290, 294, 296, 298, 300, 310, 320, 324, 328, 334, 348, 354, 364, 366, 370, 372, 376, 378, 384, 386, 390, 392, 396, 400

Offset: 1

Corinna Regina Böger, Dec 11 2016

nonn

This sequence contains all even numbers that are not in A279040 or in A273457. I have verified numerically for all even numbers 4 < 2n < 4*10^10 that a Goldbach partition with the additional condition p > sqrt(2n) exists. It is conjectured that a(n) = 2*(n+12987) for all n > 7838315. If this conjecture is true, all even numbers 2n > 15702604 have all three types of Goldbach partitions and therefore satisfy the "extended Goldbach conjecture".

			a(1) = 34 = 3 + 31 = 5 + 29 = 11 + 23 = 17 + 17. Since 3 < sqrt(17) < 5 < sqrt(34) < 11 < 17, all three types of Goldbach partitions exist for 34.

Corinna Regina Böger, Table of n, a(n) for n = 1..100000
Index entries for sequences related to Goldbach conjecture

Cf. A002373, A020481, A025018, A025019, A244408, A273457, A279040.

GoldbachRange(n,mn,mx)=forprime(p=mn,mx, if(isprime(n-p), return(1))); 0
is(n)=n%2==0 && GoldbachRange(n, 2, sqrtint(n/2)) && GoldbachRange(n, sqrtint(n/2-1)+1, sqrtint(n)) && GoldbachRange(n, sqrtint(n-1)+1, n/2) \\ Charles R Greathouse IV, Dec 16 2016

A350713 Maximum smallest prime required to generate all Goldbach partitions to 10^n.

Original entry on oeis.org

3, 19, 73, 173, 293, 523, 751, 1093, 1789, 1877, 2803, 3457, 3917, 4909, 5569, 6961, 7753, 9341

Offset: 1

Barry Cherkas, Feb 02 2022

The magnitude of the smallest prime required in a Goldbach partition of 2n is very small in comparison to the magnitude of the sum, 2n.

			The first three partitions with the smallest first member are (3,3), (3,5), and (3,7), so the smallest prime required to generate all Goldbach partitions up through 10^1 is 3.

Cf. A025018, A025019.

gp = Compile[{{n, Integer}}, Block[{p = 2}, While[! PrimeQ[n - p], p = NextPrime@p]; p]]; a[n] = 3; a[n] := Block[{k = 10^(n - 1), lmt = 10^n + 1, mx = 0}, While[k < lmt, b = gp@k; If[b > mx, mx = b]; k += 2]; mx]; (* Robert G. Wilson v, Mar 04 2022 *)

a(9)-a(18) from Robert G. Wilson v, Mar 04 2022

cp's OEIS Frontend

A245077 Largest k such that the smallest prime satisfying Goldbach's conjecture is less than or equal to (2n)^(1/k).

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A274189 Even numbers 2n that satisfy an extended Goldbach conjecture: They have at least one Goldbach partition 2n = p + q (p <= q; p, q prime) that satisfies p <= sqrt(n), at least one with sqrt(n) < p <= sqrt(2n) and at least one with p > sqrt(2n).

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PARI

A350713 Maximum smallest prime required to generate all Goldbach partitions to 10^n.

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Extensions