1,1

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),... The m-th antidiagonal of the array consists of the m elements B(m,1), B(m-1,2), B(m-2,3),...,B(1,m).

The Third Borve Conjecture states that there is an infinite number of integers m for which the m-th diagonal and m-th anti-diagonal are both zero-only.

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.......

N. Fernandez, The prime-composite array, B(m,n) and the Borve conjectures

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.

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[ Join[ Table[a[[n - i + 1, i]], {i, 1, n}], Table[a[[n + i - 1, i]], {i, 1, m - n + 1}]]] == {0}, Print[n]], {n, 1, m}]

Cf. A067677.

There is a table, see A063173 and A067681, that will work for A014617, A067677, A067681 and A063173, A063174, A063175, A063176.

Sequence in context: A066934 A137148 A045018 this_sequence A132356 A024604 A025103

Adjacent sequences: A067678 A067679 A067680 this_sequence A067682 A067683 A067684

nonn

Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 04 2002