- GOLDBACH SEPARATION FUNCTION

The underlying "Goldbach separation" data in this table was computed

by Tony D. Noe in

https://oeis.org/A155765/b155765.txt

https://oeis.org/A155764/b155764.txt

https://oeis.org/A047160

https://oeis.org/wiki/User:Jason_Kimberley/A047160

Thanks to Maximilian Hasler for pointing it out.

For N=1..61, the Nth record-high value of m is printed,

where m is the least number>=0 such that both a+m and a-m are primes, and

we keep incrementing a starting from 2 (until a new record-high m is found).

The final column is f(a) = 0.769422*ln(a)^3. This is my empirical

upper bound on m, which happens to be valid within the range of the table.

It can be too high by as much as a factor of 4.715.

[When m=0 it is infinitely too high, but I ignore that.]

N a m f(a)

1, 2, 0, 0.2562370500

2, 4, 1, 2.049893073

3, 8, 3, 6.918386629

4, 22, 9, 22.72372535

5, 46, 15, 43.18159525

6, 121, 18, 84.86824603

7, 128, 21, 87.88911839

8, 136, 27, 91.22489509

9, 146, 33, 95.23484236

10, 238, 39, 126.0861302

11, 265, 42, 133.6608183

12, 286, 45, 139.2165668

13, 341, 48, 152.6127767

14, 344, 75, 153.3014591

15, 526, 87, 189.2302088

16, 904, 93, 242.6605587

17, 1114, 117, 265.6928937

18, 1691, 120, 315.9877223

19, 1736, 135, 319.3490428

20, 1751, 138, 320.4553082

21, 1781, 168, 322.6471859

22, 2476, 183, 367.1566981

23, 3097, 210, 399.6126911

24, 3551, 228, 420.3639088

25, 5131, 300, 479.7382423

26, 8504, 333, 569.9862949

27, 10342, 369, 607.7710100

28, 18526, 393, 730.1603524

29, 22564, 453, 775.0008925

30, 24776, 525, 796.8949511

31, 40072, 621, 915.9875978

32, 68707, 720, 1063.016969

33, 125903, 810, 1246.038365

34, 174913, 846, 1353.652005

35, 181267, 1086, 1365.690833

36, 371428, 1281, 1623.109918

37, 827576, 1305, 1946.678604

38, 936118, 1515, 1999.977075

39, 1054141, 1590, 2052.240939

40, 1159864, 1617, 2094.964675

41, 1353559, 1722, 2165.244879

42, 2591107, 1794, 2477.958278

43, 3759164, 1833, 2670.035444

44, 5454062, 1851, 2871.819721

45, 5969581, 2010, 2922.274876

46, 6363983, 2064, 2958.371376

47, 6639736, 2085, 2982.466788

48, 6791669, 2112, 2995.371977

49, 7183186, 2217, 3027.501431

50, 9122719, 2352, 3167.106409

51, 10786495, 2754, 3267.468091

52, 23848807, 2784, 3771.667809

53, 27072352, 3339, 3856.745467

54, 46330868, 3429, 4231.517031

55, 57989356, 4395, 4395.000000

56, 183234439, 4488, 5299.392017

57, 297456022, 5391, 5714.632343

58, 1010526289, 5430, 6857.998184

59, 1085549026, 5877, 6929.306766

60, 1387050394, 6195, 7177.091207

61, 2117398594, 7107, 7618.514999 - --- In primenumbers@yahoogroups.com, Robert Gerbicz <robert.gerbicz@...> wrote:

> Using exhaustive search there is no better solution,

Thanks, Robert. Now Maximilian can update http://oeis.org/A217016

> so n(31)=281291010 and n(37)=10919808900.

David