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A080092
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Irregular triangle read by rows, giving prime sequences (p-1|2n) appearing in the n-th von Staudt-Clausen sum.
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7
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2, 2, 3, 2, 3, 5, 2, 3, 7, 2, 3, 5, 2, 3, 11, 2, 3, 5, 7, 13, 2, 3, 2, 3, 5, 17, 2, 3, 7, 19, 2, 3, 5, 11, 2, 3, 23, 2, 3, 5, 7, 13, 2, 3, 2, 3, 5, 29, 2, 3, 7, 11, 31, 2, 3, 5, 17, 2, 3, 2, 3, 5, 7, 13, 19, 37, 2, 3, 2, 3, 5, 11, 41, 2, 3, 7, 43, 2, 3, 5, 23, 2, 3, 47, 2, 3, 5, 7, 13, 17, 2, 3
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Comments from Gary W. Adamson & Mats O. Granvik (qntmpkt(AT)yahoo.com), Aug 09 2008 (Start) The Von Staudt-Clausen theorem has two parts: generating denominators of the B_2n and the actual values. Both operations can be demonstrated in triangles A143343 and A080092 by following the procedures outlined in [Wikipedia - Bernoulli numbers] and summarized in A143343.
A046886(n-1) = number of terms in row n.
The same terms in A143343 may be extracted from triangle A138239.
Extract primes from even numbered rows of triangle A143343 but also include "2" as row 1. The rows are thus 1, 2, 4, 6...; generating denominators of B_1, B_2, B_4,...as well as B_1, B_2, B_4,...; as two parts of the Von Staudt-Clausen theorem.
The denominator of B_12 = 2730 = (2*3*5*7*13) = A027642(12) and A002445(6).
For example, B_12 = -691/2730 = (1 - 1/2 - 1/3 - 1/5 - 1/7 - 1/13)
The second operation is the Von Staudt-Clausen representation of Bn, obtained by starting with "1" then subtracting the reciprocals of terms in each row. (Cf. A143343 for a detailed explanation of the operations). (End)
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LINKS
| Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem
Anonymous, Von Staudt-Clausen theorem, Wikipedia.
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EXAMPLE
| First few rows of the triangle are:
2;
2, 3;
2, 3, 5;
2, 3, 7;
2, 3, 5;
2, 3, 11;
2, 3, 5, 7, 13;
2, 3;
...
Sum for n=1 is 1/2+1/3, so terms are 2, 3. Sum for n=2 is 1/2+1/3+1/5, so terms are 2, 3, 5. Etc.
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MATHEMATICA
| row[n_] := Select[ Prime /@ Range[n+1], Divisible[2n, # - 1] &]; Flatten[Table[row[n], {n, 0, 25}]] (* From Jean-François Alcover, Oct 12 2011 *)
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CROSSREFS
| Cf. A000146, A143343, A138239, A002445, A027642.
Sequence in context: A022467 A169614 A037126 * A164738 A126225 A056160
Adjacent sequences: A080089 A080090 A080091 * A080093 A080094 A080095
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KEYWORD
| nonn,easy,nice,tabf
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com), Jan 27, 2003
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EXTENSIONS
| Edited by N. J. A. Sloane, Nov 01 2009 at the suggestion of R. J. Mathar
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