|
| |
|
|
A067677
|
|
Diagonals of the prime-composite array, B(m,n) which are zeros from the Second Borve Conjecture.
|
|
2
|
|
|
|
8, 12, 26, 35, 38, 53, 66, 73, 77, 90, 121, 126, 129, 144, 150, 195, 208, 223, 245, 258, 260, 270, 280, 308, 355, 379, 388, 395, 413, 419, 431, 486, 497, 502, 510, 560, 650, 665, 694, 727, 736, 753, 758, 779, 789, 792, 820
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Let c(m) be the m-th composite and p(n) be the n-th prime. The prime-composite array, B, is defined such that each element B(m,n) is the highest power of p(n) that is contained within c(m). Diagonals can also be specified, where the m-th diagonal consists of the infinite number of elements B(m,1), B(m+1,2), B(m+2,3), ...
Diagonals can also be specified, where the m-th diagonal consists of the infinite number of elements B(m,1), B(m+1,2), B(m+2,3), ...
The Second Borve Conjecture states that there are infinitely many zero-only diagonals.
The prime-composite array begins:
1 2 3 4 5 6 7 8 (n)
(2) (3) (5) (7) (11) (13) (17) (19) (p_n)
1 (4) 2 0 0 0 0 0 0 0 ...
2 (6) 1 1 0 0 0 0 0 0 ...
3 (8) 3 0 0 0 0 0 0 0 ...
4 (9) 0 2 0 0 0 0 0 0 ...
5 (10) 1 0 1 0 0 0 0 0 ...
6 (12) 2 1 0 0 0 0 0 0 ...
7 (14) 1 0 0 1 0 0 0 0 ...
8 (15) 0 1 1 0 0 0 0 0 ...
9 (16) 4 0 0 0 0 0 0 0 ...
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Thus each composite has its own row, consisting of the indices of its prime factors. For example, the 10th composite is 18 and 18 = 2^1 * 3^2 * 5^0 * 7^0 * 11^0 * ..., so the 10th row reads: 1, 2, 0, 0, 0, .... Similarly, B(6,2) = 1 because c(6) = 12, p(2) = 3 and the highest power of 3 contained within 12 is 3^1 = 3. And B(34,3) = 2 because c(34) = 50, p(3) = 5 and the highest power of 5 contained within 50 is 5^2 = 25.
|
|
|
MATHEMATICA
|
Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; m = 750; a = Table[0, {m}, {m}]; Do[b = Transpose[ FactorInteger[ Composite[n]]]; a[[n, PrimePi[First[b]]]] = Last[b], {n, 1, m}]; Do[ If[ Union[ Table[ a[[n + i - 1, i]], {i, 1, m - n + 1} ]] == {0}, Print[n]], {n, 1, m}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
