%I A067677
%S A067677 8,12,26,35,38,53,66,73,77,90,121,126,129,144,150,195,208,223,245,258,
%T A067677 260,270,280,308,355,379,388,395,413,419,431,486,497,502,510,560,650,
%U A067677 665,694,727,736,753,758,779,789,792,820
%N A067677 Diagonals of the prime-composite array, B(m,n) which are zeros from the 
               Second Borve Conjecture.
%C A067677 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). Diagonals can also be 
               specified, where the m-th diagonal consists of the infinite number 
               of elements B(m,1), B(m+1,2), B(m+2,3),...
%C A067677 Diagonals can also be specified, where the m-th diagonal consists of 
               the infinite number of elements B(m,1), B(m+1,2), B(m+2,3),...
%C A067677 The Second Borve Conjecture states that there is an infinite number of 
               zero-only diagonals.
%C A067677 The prime-composite array begins:
%C A067677 ...... .1....2....3....4....5....6....7....8....(n)
%C A067677 ...... (2)...(3)..(5)..(7).(11).(13).(17).(19).(p_n)
%C A067677 1 .(4) .2....0....0....0....0....0....0....0.......
%C A067677 2 .(6) .1....1....0....0....0....0....0....0.......
%C A067677 3 .(8) .3....0....0....0....0....0....0....0.......
%C A067677 4 .(9) .0....2....0....0....0....0....0....0.......
%C A067677 5 (10) .1....0....1....0....0....0....0....0.......
%C A067677 6 (12) .2....1....0....0....0....0....0....0.......
%C A067677 7 (14) .1....0....0....1....0....0....0....0.......
%C A067677 8 (15) .0....1....1....0....0....0....0....0.......
%C A067677 9 (16) .4....0....0....0....0....0....0....0.......
%H A067677 N. Fernandez, <a href="http://www.borve.org/primeness/pcarray.html">The 
               prime-composite array, B(m,n) and the Borve conjectures</a>
%e A067677 Thus 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 A067677 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, m - n + 1} ]] == 
               {0}, Print[n]], {n, 1, m}]
%Y A067677 Cf. A067681.
%Y A067677 Sequence in context: A054735 A162691 A077566 this_sequence A045523 A006983 
               A072327
%Y A067677 Adjacent sequences: A067674 A067675 A067676 this_sequence A067678 A067679 
               A067680
%K A067677 nonn
%O A067677 1,1
%A A067677 Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 04 2002