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A059861
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Product(p(i)-2), i=2,3...n where p(i) = i-th prime.
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5
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1, 1, 3, 15, 135, 1485, 22275, 378675, 7952175, 214708725, 6226553025, 217929355875, 8499244879125, 348469040044125, 15681106801985625, 799736446901266875, 45584977473372211875, 2689513670928960500625
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,3
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COMMENTS
| Arises in Hardy-Littlewood k-tuple conjecture. Also a(n) is the exact number of d=2 and also d=4 differences in dRRS[modulus=nth primorial]; see A049296 (dRRS[m]=set of first differences of reduced residue system modulo m).
For n>1 this is the determinant of the (n-1) X (n-1) matrix whose diagonal is A006093(n) = {1, 2, 4, 6, 10, 12, 16, 18..} = the first primes minus 1 and all other elements are 1's. The determinant begins: / (2-1) 1 1 1 1 1 1 ... / 1 (3-1) 1 1 1 1 1 ... / 1 1 (5-1) 1 1 1 1 ... / 1 1 1 (7-1) 1 1 1 ... / 1 1 1 1 (11-1) 1 1 ... / 1 1 1 1 1 (13-1) 1 ... - Alexander Adamchuk (alex(AT)kolmogorov.com), May 21 2006
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 21 2009: (Start)
Equals (-1)^n * (1, 1, 1, 3, 15,...) dot (1, -2, 4, -6, 10,...).
a(6) = 135 = (1, 1, 1, 3, 15) dot (1, -2, 4, -6, 10) = (1, -2, 4, -18, 150). (End)
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REFERENCES
| S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, Sections A8, A1.
G. H. Hardy and J. E. Littlewood, "Partitio Numerorum III", Acta Math. 44 (1922) 1-70.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954.
G. Polya, Heuristic reasoning in the theory of numbers Am. Math. Monthly, 66 (1959), 375-384.
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LINKS
| C. K. Caldwell, Prime k-tuple Conjecture
S. R. Finch, Hardy-Littlewood Constants
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
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FORMULA
| a(n) = Det[ DiagonalMatrix[ Table[ Prime[i-1] - 2, {i, 2, n} ] ] + 1 ] for n>1. - Alexander Adamchuk (alex(AT)kolmogorov.com), May 21 2006
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EXAMPLE
| n=4, a(4)=1*(3-2)*(5-2)*(7-2)=15. 48 first terms of A049296 give one complete period of dRRS[210], in which 15 d=2, 15 d=4 and 18 larger differences occur. For n=1, 2, ..., 5 in the periods of length {1, 2, 8, 48, 480, ..} [see A005867] the number of d=2 and also d=4 differences is {1, 1, 3, 15, 135, ..}
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MATHEMATICA
| Table[ Det[ DiagonalMatrix[ Table[ Prime[i-1] - 2, {i, 2, n} ] ] + 1 ], {n, 2, 20} ] - Alexander Adamchuk (alex(AT)kolmogorov.com), May 21 2006
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CROSSREFS
| Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865, A040976.
Cf. A067549, A006093.
Sequence in context: A117694 A108210 A006717 * A030539 A028362 A195764
Adjacent sequences: A059858 A059859 A059860 * A059862 A059863 A059864
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KEYWORD
| nonn
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AUTHOR
| Labos E. (labos(AT)ana.sote.hu), Feb 28 2001
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