[seqfan] Re: [SPAM?] b-Zahlen

[Peter]

Hi David,

using Mathematica I get slightly different b-numbers. For instance: 308
is in b_4, not in b_5.

The sequences start as follows:

{k = 1,{4,6,8,10,12,14,16,18,20,22}}
{k = 2,{122,128,148,190,208,250,292,302,326,332}}
{k = 3,{98,220,346,368,398,458,518,586,640,692}}
{k = 4,{308,854,908,968,1138,1150,1202,1258,1354,1408}}
{k = 5,{488,556,1144,1382,1532,1774,1922,2188,2512,2618}}
{k = 6,{962,1118,1856,2432,3076,3526,3754,3794,4322,4418}}
{k = 7,{1768,2078,4508,4712,5978,6758,7292,7862,7868,8156}}
{k = 8,{3818,3848,4618,7928,8888,9124,9266,9512,9922,10232}}
{k = 9,{1928,2438,6008,11042,11438,12778,15788,17708,18968,19246}}
{k = 10,{992,9488,10622,12212,13058,18782,18908,19828,22916,23266}}
{k = 11,{14198,26822,28052,28096,30518,35198,36692,37768,40064,40948}}
{k = 12,{2642,10268,11642,13148,22606,35618,44588,52102,53266,55556}}
{k = 13,{5372,9602,21368,24886,34096,41308,55172,67700,71144,73498}}
{k = 14,{7426,14678,23456,32702,42308,50312,69428,75304,77018,88786}}
{k = 15,{9596,12886,13562,16502,23426,43532,55142,60062,127802,138128}}
{k = 16,{64838,110108,120602,142502,167072,188116,193718,194344,197048}}
{k = 17,{54244,57188,93452,96052,111692,152288}}
{k = 18,{48002,111722,194302,194452,198758}}
{k = 19,{22832,66158,134972,194386}}
{k = 20,{100768,131666}}
{k = 21,{103738,137708,181388}}
{k = 22,{63274,84116,113672}}
{k = 23,{194470}}
{k = 24,{194428}}
{k = 25,{128168}}
{k = 26,{180596}}

I used a simple and inefficient testing-function


test[n_, k_] := Block[{pn = PrimePi[n-2]}, MatchQ[ PrimeQ[n -
Prime[Range[pn - k + 1, pn]]], {True, False ...}]];

where n is already the even number to test (the "2 n" in your post).


and generated a table of all b_k for k <=30 which are less than or equal
to 2*10^5 by entering

tbl = ParallelTable[ Pick[#, test[#, k]]& [Range[Prime[k] + 2 + Boole[k
>=2 ], 2 * 10^5, 2]], {k, 1, 30}];


I take the liberty to send you a compressed .CSV-file because it is not
too large (176 kB,  ~4.3 MB uncompressed).


Kind regards,

Peter



Am 07.12.2018 um 18:10 schrieb David Sycamore via SeqFan:
> ...

> Let b_k denote the even numbers 2n such that the greatest prime q < 2n-1 having the property that 2n-q is prime is the k-th prime below (less than) 2n-1. Then A244207 is the sequence of even numbers 2n such that k > 1. 
>
> And we have the « b_k-Zahlen » sequences:
>
> b_1 : A005843 less {0,2} and all numbers in A244207.
>
> And then the following subsequences  of A244207 : 
>
> b_2 : 122,128,148,190,208,220,250,
> 292,302,326,332,410,418,430......
>
> b_3 : 98, 346, 368, 398, 458, 518, 586, 640, 692.....
>
> b_4 : 854, 908.....
>
> b_5 : 308, 488, 556....
>
> This is as far as I’ve got;  nothing yet for k >= 6. 
>
> Also: a(n) = smallest number in sequence b_n :
>
> 4, 122, 98, 854, 308...
>
> None of the above are in Oeis, although the b_4 numbers seem to appear in A244408, A279040 and A038433.
>
> I doubt  very much if any of this is of any use, but is it of interest to Oeis ? 
>
> Regards
>
> David.
>
>
>
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