OFFSET
1,3
COMMENTS
For each k except for k=2 (whose value in row 1 is already at the maximum possible for that column), the values in the column, as n increases, increase to a maximum before beginning to descend toward zero. This behavior of the numbers in a given column can be viewed as the net result of two competing effects as n increases.
First, each time n is incremented, the smallest prime that can be a proper divisor of one of the two nearest neighbors of a number m*prime(n)# (thus preventing one of those two nearest neighbors from being a prime) loses its ability to divide either of those neighbors. E.g., at n=1, the candidate numbers to be tested to determine whether they are the average of a twin prime pair are numbers of the form m*prime(1)# = m*2# = m*2, i.e., even numbers, and two out of every three consecutive even numbers are prevented from being the average of a twin prime pair because one of the even number's two neighbors (m*2 -+ 1) is a proper multiple of 3. E.g., at k=4, the 4-bit numbers m are 8 through 15, but of those, m = 8, 10, 11, 13, and 14 cannot yield m*2 as the average of a twin prime pair because m*2 - 1 or m*2 + 1 is a proper multiple of 3, hence not a prime. But when we move to n=2, the candidates to be tested to determine whether they are the average of a twin prime pair are now numbers of the form m*prime(2)# = m*3# = m*2*3 = m*6, and no number of the form m*6 that fails to be the average of a twin prime pair does so because it has a neighbor that is divisible by 3.
Second, each time n is incremented, the numbers to be tested to determine whether they are the average of a twin prime pair get larger by a factor of prime(n). E.g., at n=4, the candidates are numbers of the form m*prime(4)# = m*7# = m*2*3*5*7 = m*210, but at n=5, the candidates are numbers of the form m*prime(5)# = m*11# = m*2*3*5*7*11 = m*2310. For a given set of numbers m (e.g., for k=10, the numbers m=512..1023), the products m*2310 may be less likely to be the average of twin primes than the smaller products m*210 because the density of primes in the vicinity of a number x decreases as x increases.
EXAMPLE
T(5,4) = 2 because there are 2 4-bit numbers m such that m*2*3*5*7*11 = m*2310 is the average of a twin prime pair:
1011_2 * 2*3*5*7*11 = 11*2310 = 25410 (the average of (25409, 25411)) and
1111_2 * 2*3*5*7*11 = 15*2310 = 34650 (the average of (34649, 34651)).
The table begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
---+----------------------------------------------------------------
1 | 0 2 1 2 2 3 7 7 12 26 45 70 113 215 355 666 1153 2071 ...
2 | 1 2 2 2 5 8 9 20 32 63 96 165 284 515 922 1624 2916 5422 ...
3 | 1 1 2 3 4 6 15 28 35 60 113 214 349 643 1181 2192 3974 7031 ...
4 | 0 1 1 2 4 8 5 22 40 62 123 205 375 717 1274 2295 4256 7857 ...
5 | 1 0 2 2 2 5 11 22 28 55 107 168 354 606 1168 2160 3974 7462 ...
6 | 0 0 1 5 3 5 9 12 26 41 82 158 263 579 1079 1954 3641 7073 ...
7 | 0 0 0 1 2 2 7 15 17 40 73 137 249 498 902 1771 3276 6255 ...
8 | 0 0 0 1 2 2 7 10 20 32 62 140 226 476 776 1530 2909 5522 ...
9 | 0 0 1 1 5 2 5 9 11 20 56 115 211 369 737 1322 2590 4859 ...
10 | 0 0 0 0 1 2 4 8 14 21 46 86 186 315 594 1212 2249 4332 ...
11 | 0 0 0 0 1 2 2 8 11 20 35 76 152 268 537 1067 2001 3779 ...
12 | 0 0 1 0 1 1 3 5 13 12 30 55 125 238 452 925 1776 3454 ...
13 | 0 0 0 0 0 0 3 3 8 20 23 56 119 211 414 799 1519 2934 ...
14 | 0 0 0 0 1 1 0 5 3 15 25 44 107 214 365 725 1322 2673 ...
15 | 0 0 0 0 0 2 0 6 5 15 19 53 85 162 302 622 1303 2398 ...
16 | 0 0 0 0 1 0 1 4 3 13 21 43 87 156 297 557 1090 2134 ...
17 | 0 0 0 1 1 1 3 1 4 13 17 37 71 134 261 530 955 1893 ...
18 | 0 0 0 1 0 0 1 1 4 11 15 29 51 118 243 480 920 1752 ...
19 | 0 0 0 0 0 0 0 1 3 14 12 33 72 120 220 433 860 1613 ...
...
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Jon E. Schoenfield, Nov 28 2023
STATUS
approved