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Search: id:A067677
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| A067677 |
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Diagonals of the prime-composite array, B(m,n) which are zeros from the Second Borve Conjecture. |
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+0
3
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| 8, 12, 26, 35, 38, 53, 66, 73, 77, 90, 121, 126, 129, 144, 150, 195, 208, 223, 245, 258, 260, 270, 280, 308, 355, 379, 388, 395, 413, 419, 431, 486, 497, 502, 510, 560, 650, 665, 694, 727, 736, 753, 758, 779, 789, 792, 820
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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),...
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 Second Borve Conjecture states that there is an infinite number of zero-only diagonals.
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
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N. Fernandez, The prime-composite array, B(m,n) and the Borve conjectures
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EXAMPLE
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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
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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}]
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CROSSREFS
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Cf. A067681.
Sequence in context: A054735 A162691 A077566 this_sequence A045523 A006983 A072327
Adjacent sequences: A067674 A067675 A067676 this_sequence A067678 A067679 A067680
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KEYWORD
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nonn
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AUTHOR
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Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 04 2002
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