

A067677


Diagonals of the primecomposite 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
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OFFSET

1,1


COMMENTS

Let c(m) be the mth composite and p(n) be the nth prime. The primecomposite 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 mth 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 mth 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 zeroonly diagonals.
The primecomposite 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



