A067276
Determinant of n X n matrix containing the first n^2 primes in increasing order.
Original entry on oeis.org
2, -1, -78, 880, -4656, -14304, -423936, 8342720, 711956736, -615707136, 21057138688, -4663930678272, 211912980656128, -9178450735677440, 40005919124799488, 83013253447139328, -8525111273818357760, -800258888289188708352, -15170733077495639179264
Offset: 1
a(3) = -78 because det[[2,7,17],[3,11,19],[5,13,23]] = -78 (= det[[2,3,5],[7,11,13],[17,19,23]], the determinant of the transpose.).
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[ Determinant( Matrix(n, n, [ NthPrime(k): k in [1..n^2] ]) ): n in [1..19] ]; // Klaus Brockhaus, May 12 2010
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seq(LinearAlgebra:-Determinant(Matrix(n,n,(i,j) -> ithprime(n*(i-1)+j))),n=1..20); # Robert Israel, Jul 12 2017
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Table[ Det[ Partition[ Array[Prime, n^2], n]], {n, 19}] (* Robert G. Wilson v, May 26 2006 *)
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for(n=1,20,k=0; m=matrix(n,n,x,y, prime(k=k+1)); print1(matdet(m), ", ")) /* The matrix initialization command above fills columns first: Variables (such as) x and y take on values 1 through n for rows and columns, respectively, with x changing more rapidly and they must be specified even though the 5th argument is not an explicit function of them here. */
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from sympy.matrices import Matrix
from sympy import sieve
def a(n):
sieve.extend_to_no(n**2)
return Matrix(n, n, sieve[1:n**2+1]).det()
print([a(n) for n in range(1, 20)]) # Indranil Ghosh, Jul 31 2017
A059861
a(n) = Product_{i=2..n} (prime(i) - 2).
Original entry on oeis.org
1, 1, 3, 15, 135, 1485, 22275, 378675, 7952175, 214708725, 6226553025, 217929355875, 8499244879125, 348469040044125, 15681106801985625, 799736446901266875, 45584977473372211875, 2689513670928960500625
Offset: 1
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, ..}
- 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.
- 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.
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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 *)
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a(n) = prod(i=2, n, prime(i)-2); \\ Michel Marcus, Apr 16 2017
A330967
a(n) is the determinant of the matrix with elements gcd(i,j) for 2 <= i,j <= n.
Original entry on oeis.org
2, 5, 10, 44, 104, 656, 2624, 15744, 67584, 694272, 2777088, 34062336, 213221376, 1758855168, 14070841344, 228530847744, 1371185086464, 25007480635392, 200059845083136, 2447683608379392, 25040421692375040, 556525133318062080, 4452201066544496640, 89044021330889932800
Offset: 2
A001088 gives the determinants for gcd(i,j), 1 <= i,j <= n.
A067549 gives the determinants for gcd(i-th prime, j-th prime), 1 <= i,j <= n.
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Table[Det[Table[GCD[i, j], {i, 2, n}, {j, 2, n}]], {n, 2, 25}]
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a(n)={matdet(matrix(n-1, n-1, i, j, gcd(i+1,j+1)))} \\ Andrew Howroyd, Jan 07 2020
Showing 1-3 of 3 results.
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