A213930 - OEIS

%I #27 May 01 2013 10:27:36

%S 2,8,7,7,1,35,40,44,15,16,7,7,0,1,1,205,202,299,101,119,105,54,33,40,

%T 15,16,15,3,5,11,1,2,1,1224,1215,1940,773,916,964,484,339,514,238,223,

%U 206,88,98,146,32,33,54,19,28,19,5,4,3,5

%N Table of frequencies of gaps of size 2d between consecutive primes below 10^n, n >= 1; d = 1,2,...,A213949(n).

%C Sum of elements in line n is Pi(10^n)-2. Column d is the sequence of the numbers of gaps of size 2d between consecutive primes up to 10^n. For example, column 1 is A007508, and column 2 is A093737. Column 3 corresponds to the jumping champion 6. Column 15 corresponds to the next champion 30. It is interesting that local maximums appear in the beginning of this column, 11 in line 4, and 146 in line 5.

%H Washington Bomfim, <a href="/A213930/b213930.txt">Rows n = 1..13, flattened</a>

%H A. Odlyzko, M. Rubinstein and M. Wolf, <a href="http://www.dtc.umn.edu/~odlyzko/doc/jumping.champions.pdf">Jumping Champions</a>

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>

%e Table begins

%e    2

%e    8    7    7   1

%e   35   40   44  15  16   7   7   0   1   1

%e   205  202  299 101 119 105  54  33  40  15  16  15  3  5  11  1  2  1

%e 1224 1215 1940 773 916 964 484 339 514 238 223 206 88 98 146 32 33 54 19 28...

%t Table[t2 = Sort[Tally[Table[Prime[k + 1] - Prime[k], {k, 2, PrimePi[10^n] - 1}]]]; maxDiff = t2[[-1, 1]]/2; t3 = Table[0, {k, maxDiff}];Do[t3[[t2[[i, 1]]/2]] = t2[[i, 2]], {i, Length[t2]}]; t3, {n, 5}] (* _T. D. Noe_, Jun 25 2012 *)

%Y Cf. A038460, A000720, A007508, A093737, A213949 (row lengths).

%K tabf,nonn,nice

%O 1,1

%A _Washington Bomfim_, Jun 24 2012