%I #19 Feb 21 2022 02:23:00

%S 4,8,12,23,30,35,46,49,70,73,88,97,102,106,118,123,146,162,167,171,

%T 195,205,236,240,242,245,254,270,272,290,292,297,320,325,332,342,355,

%U 365,374,444,453,502,508,523,532,578,585,596,599,609,634,645,663,677,687

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

%C 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.

%H N. Fernandez, <a href="http://www.borve.org/primeness/pcarray.html">The prime-composite array, B(m,n) and the Borve conjectures</a>

%e 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.

%t 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}]

%Y Cf. A067681.

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Feb 04 2002