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%I #71 Nov 14 2023 04:34:31
%S 1,1,3,15,135,1485,22275,378675,7952175,214708725,6226553025,
%T 217929355875,8499244879125,348469040044125,15681106801985625,
%U 799736446901266875,45584977473372211875,2689513670928960500625
%N a(n) = Product_{i=2..n} (prime(i) - 2).
%C 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=n-th primorial]; see A049296 (dRRS[m]=set of first differences of reduced residue system modulo m).
%C 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_, May 21 2006
%C From _Gary W. Adamson_, Apr 21 2009: (Start)
%C Equals (-1)^n * (1, 1, 1, 3, 15, ...) dot (1, -2, 4, -6, 10, ...).
%C a(6) = 135 = (1, 1, 1, 3, 15) dot (1, -2, 4, -6, 10) = (1, -2, 4, -18, 150). (End)
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
%D R. K. Guy, Unsolved Problems in Number Theory, Sections A8, A1.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
%D G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954.
%H A.H.M. Smeets, <a href="/A059861/b059861.txt">Table of n, a(n) for n = 1..100</a>
%H Steven Brown, <a href="https://arxiv.org/abs/2311.06873">Distance between consecutive elements of the multiplicative group of integers modulo n</a>, arXiv:2311.06873 [math.NT], 2023. See Table 1 p. 25.
%H C. K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=PrimeKtupleConjecture">Prime k-tuple Conjecture</a>
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/hrdyltl/hrdyltl.html">Hardy-Littlewood Constants </a> [Broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010614100031/http://www.mathsoft.com/asolve/constant/hrdyltl/hrdyltl.html">Hardy-Littlewood Constants </a> [From the Wayback machine]
%H G. H. Hardy and J. E. Littlewood, <a href="https://dx.doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a> [Cached copy]
%H G. Polya, <a href="http://www.jstor.org/stable/2308748">Heuristic reasoning in the theory of numbers</a>, Am. Math. Monthly, 66 (1959), 375-384.
%F a(n) = Det[ DiagonalMatrix[ Table[ Prime[i-1] - 2, {i, 2, n} ] ] + 1 ] for n>1. - _Alexander Adamchuk_, May 21 2006
%F a(n) = a(n-1) * (A000040(n) - 2) for n > 1. - _A.H.M. Smeets_, Dec 14 2019
%F a(n) = |{r | 0 <= r < primorial(n) and gcd(r, primorial(n)) = 1 and gcd(r + 2, primorial(n)) = 1}|. - _Greg Tener_, Oct 22 2021
%e 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, ..}
%t Table[ Det[ DiagonalMatrix[ Table[ Prime[i-1] - 2, {i, 2, n} ] ] + 1 ], {n, 2, 20} ] (* _Alexander Adamchuk_, May 21 2006 *)
%t Table[Product[Prime@k - 2, {k, 2, n}], {n, 1, 18}] (* _Harlan J. Brothers_, Jul 02 2018 *)
%t a[1] = 1; a[n_] := a[n] = a[n - 1] (Prime[n] - 2);
%t Table[a[n], {n, 18}] (* _Harlan J. Brothers_, Jul 02 2018 *)
%t Join[{1},FoldList[Times,Prime[Range[2,20]]-2]] (* _Harvey P. Dale_, Apr 19 2023 *)
%o (PARI) a(n) = prod(i=2, n, prime(i)-2); \\ _Michel Marcus_, Apr 16 2017
%Y Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865, A040976.
%Y Cf. A067549, A006093.
%K nonn
%O 1,3
%A _Labos Elemer_, Feb 28 2001
%E Offset corrected by _A.H.M. Smeets_, Dec 14 2019
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