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A067681
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Diagonals and antidiagonals of the prime-composite array, B(m,n) which are zeros from the Third Borve Conjecture.
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4
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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
(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). 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 is an infinite number of integers m for which the m-th diagonal and m-th anti-diagonal 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.......
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LINKS
| N. Fernandez, The prime-composite array, B(m,n) and the Borve conjectures
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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.
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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[ 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}]
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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: A066934 A137148 A045018 * A132356 A024604 A025103
Adjacent sequences: A067678 A067679 A067680 * A067682 A067683 A067684
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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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