

A014617


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


1



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
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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). The mth antidiagonal of the array consists of the m elements B(m,1), B(m1,2), B(m2,3),...,B(1,m). The First Borve Conjecture states that there is an infinite number of zeroonly antidiagonals.


LINKS

Table of n, a(n) for n=1..55.
N. Fernandez, The primecomposite 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

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



