A014617 Antidiagonals of the prime-composite array B(m,n) (see A067681) that are zeros from the first Borve conjecture.

4, 8, 12, 23, 30, 35, 46, 49, 70, 73, 88, 97, 102, 106, 118, 123, 146, 162, 167, 171, 195, 205, 236, 240, 242, 245, 254, 270, 272, 290, 292, 297, 320, 325, 332, 342, 355, 365, 374, 444, 453, 502, 508, 523, 532, 578, 585, 596, 599, 609, 634, 645, 663, 677, 687

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). 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 are infinitely many zero-only antidiagonals.

Links

Table of n, a(n) for n=1..55.

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

Example

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, n} ]] == {0}, Print[n]], {n, 1, m}]

Crossrefs

Cf. A067681.

Sequence in context: A092108 A015781 A130643 * A239053 A272708 A157416

Adjacent sequences: A014614 A014615 A014616 * A014618 A014619 A014620

Keyword

nonn

Author

Robert G. Wilson v, Feb 04 2002

Status

approved