

A067681


Diagonals and antidiagonals of the primecomposite array, B(m,n) which are zeros from the Third Borve Conjecture.


4



8, 12, 35, 73, 195, 245, 270, 355, 502, 885, 890, 1069, 1096, 1228, 1403, 1451, 1639, 2082, 2087, 2131, 2142, 2376, 2418, 2524, 2582, 2683, 2953, 3236, 3262, 3267, 3289, 3392, 3587, 3642, 4119, 4161
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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). Diagonals can also be specified, where the mth diagonal consists of the infinite number of elements B(m,1), B(m+1,2), B(m+2,3), ... The mth antidiagonal of the array consists of the m elements B(m,1), B(m1,2), B(m2,3), ..., B(1,m).
The Third Borve Conjecture states that there is an infinite number of integers m for which the mth diagonal and mth antidiagonal are both zeroonly.
The primecomposite array begins:
. .... .1....2....3....4....5....6....7....8....(n)
. .... (2)..(3)..(5)..(7).(11).(13).(17).(19)..(p_n)
1 .(4) .2....0....0....0....0....0....0....0.......
2 .(6) .1....1....0....0....0....0....0....0.......
3 .(8) .3....0....0....0....0....0....0....0.......
4 .(9) .0....2....0....0....0....0....0....0.......
5 (10) .1....0....1....0....0....0....0....0.......
6 (12) .2....1....0....0....0....0....0....0.......
7 (14) .1....0....0....1....0....0....0....0.......
8 (15) .0....1....1....0....0....0....0....0.......
9 (16) .4....0....0....0....0....0....0....0.......


LINKS

Table of n, a(n) for n=1..36.
N. Fernandez, The primecomposite array, B(m,n) and the Borve conjectures


EXAMPLE

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.


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[ Join[ Table[a[[n  i + 1, i]], {i, 1, n}], Table[a[[n + i  1, i]], {i, 1, m  n + 1}]]] == {0}, Print[n]], {n, 1, m}]


CROSSREFS

Cf. A067677.
There is a table, see A063173 and A067681, that will work for A014617, A067677, A067681 and A063173, A063174, A063175, A063176.
Sequence in context: A137148 A211778 A045018 * A132356 A282943 A024604
Adjacent sequences: A067678 A067679 A067680 * A067682 A067683 A067684


KEYWORD

nonn


AUTHOR

Robert G. Wilson v, Feb 04 2002


STATUS

approved



