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 A067677 Diagonals of the prime-composite array, B(m,n) which are zeros from the Second Borve Conjecture. 2
 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; refs; listen; history; text; internal format)
 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), ... 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 are infinitely many 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 ... LINKS Table of n, a(n) for n=1..47. 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[ Table[ a[[n + i - 1, i]], {i, 1, m - n + 1} ]] == {0}, Print[n]], {n, 1, m}] CROSSREFS Cf. A067681. Sequence in context: A368130 A367894 A210982 * A045523 A006983 A283148 Adjacent sequences: A067674 A067675 A067676 * A067678 A067679 A067680 KEYWORD nonn AUTHOR Robert G. Wilson v, Feb 04 2002 STATUS approved

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Last modified July 20 14:35 EDT 2024. Contains 374445 sequences. (Running on oeis4.)