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A014617
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Anti-diagonals of the prime-composite array B(m,n) (see A067681) which are zeros from the first Borve conjecture.
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1
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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 is an infinite number of zero-only antidiagonals.
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LINKS
| N. Fernandez, The prime-composite array, B(m,n) and the Borve conjectures
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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.
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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}]
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CROSSREFS
| Sequence in context: A092108 A015781 A130643 * A157416 A059992 A050570
Adjacent sequences: A014614 A014615 A014616 * A014618 A014619 A014620
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KEYWORD
| nonn
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AUTHOR
| Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 04 2002
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