A059861 a(n) = Product_{i=2..n} (prime(i) - 2).

1, 1, 3, 15, 135, 1485, 22275, 378675, 7952175, 214708725, 6226553025, 217929355875, 8499244879125, 348469040044125, 15681106801985625, 799736446901266875, 45584977473372211875, 2689513670928960500625

Offset

1,3

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=n-th 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, May 21 2006

From Gary W. Adamson, 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)

References

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

R. K. Guy, Unsolved Problems in Number Theory, Sections A8, A1.

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.

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..100

Steven Brown, Distance between consecutive elements of the multiplicative group of integers modulo n, arXiv:2311.06873 [math.NT], 2023. See Table 1 p. 25.

C. K. Caldwell, Prime k-tuple Conjecture

Steven R. Finch, Hardy-Littlewood Constants [Broken link]

Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]

G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.

G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly, 66 (1959), 375-384.

Formula

a(n) = Det[ DiagonalMatrix[ Table[ Prime[i-1] - 2, {i, 2, n} ] ] + 1 ] for n>1. - Alexander Adamchuk, May 21 2006

a(n) = a(n-1) * (A000040(n) - 2) for n > 1. - A.H.M. Smeets, Dec 14 2019

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

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, ..}

Mathematica

Table[ Det[ DiagonalMatrix[ Table[ Prime[i-1] - 2, {i, 2, n} ] ] + 1 ], {n, 2, 20} ] (* Alexander Adamchuk, May 21 2006 *)
Table[Product[Prime@k - 2, {k, 2, n}], {n, 1, 18}] (* Harlan J. Brothers, Jul 02 2018 *)
a[1] = 1; a[n_] := a[n] = a[n - 1] (Prime[n] - 2);
Table[a[n], {n, 18}]  (* Harlan J. Brothers, Jul 02 2018 *)
Join[{1}, FoldList[Times, Prime[Range[2, 20]]-2]] (* Harvey P. Dale, Apr 19 2023 *)

Prog

(PARI) a(n) = prod(i=2, n, prime(i)-2); \\ Michel Marcus, Apr 16 2017

Crossrefs

Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865, A040976.

Cf. A067549, A006093.

Sequence in context: A222263 A246804 A230166 * A348420 A232699 A030539

Adjacent sequences: A059858 A059859 A059860 * A059862 A059863 A059864

Keyword

nonn

Author

Labos Elemer, Feb 28 2001

Extensions

Offset corrected by A.H.M. Smeets, Dec 14 2019

Status

approved