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). 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 First Borve Conjecture states that there is an infinite number of zero-only antidiagonals.

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

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

Sequence in context: A092108 A015781 A130643 this_sequence A059992 A050570 A102110

Adjacent sequences: A014614 A014615 A014616 this_sequence A014618 A014619 A014620

nonn

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