A367739 Table read by ascending antidiagonals: T(n,k) is the number of k-bit numbers m such that m*prime(n)# is the average of a twin prime pair, where prime(n)# is the n-th primorial A002110(n).

0, 1, 2, 1, 2, 1, 0, 1, 2, 2, 1, 1, 2, 2, 2, 0, 0, 1, 3, 5, 3, 0, 0, 2, 2, 4, 8, 7, 0, 0, 1, 2, 4, 6, 9, 7, 0, 0, 0, 5, 2, 8, 15, 20, 12, 0, 0, 0, 1, 3, 5, 5, 28, 32, 26, 0, 0, 1, 1, 2, 5, 11, 22, 35, 63, 45, 0, 0, 0, 1, 2, 2, 9, 22, 40, 60, 96, 70, 0, 0, 0, 0, 5, 2, 7, 12, 28, 62, 113, 165, 113

Offset: 1

Author

Jon E. Schoenfield, Nov 28 2023

Keywords

• nonn
• tabl

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.

Examples

			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

Cf. A014574, A095017.