A067681 Diagonals and antidiagonals of the prime-composite array, B(m,n) which are zeros from the Third Borve Conjecture.

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

Offset

1,1

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). 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), ... 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 Third Borve Conjecture states that there are infinitely many integers m for which the m-th diagonal and m-th antidiagonal are both zero-only.

The prime-composite 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 prime-composite 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 A341781 A282943

Adjacent sequences: A067678 A067679 A067680 * A067682 A067683 A067684

Keyword

nonn

Author

Robert G. Wilson v, Feb 04 2002

Status

approved