A117301 -id:A117301 - cp's OEIS frontend

A118799 Determinants of 4 X 4 matrices of 16 consecutive primes.

880, -448, -1472, -240, 2480, -1352, -4128, -96, 2736, -2520, 120, 1080, 4288, 4880, 4600, 13368, 7056, 14560, 2960, 13320, 0, 24864, -11096, -24264, 0, -9168, -2128, -15792, 0, 18120, -5248, 6384, -21840, -38776, -20480, 20176, -72896, -69200, 40080, -37632

Offset: 1

Jonathan Vos Post, May 23 2006

4 X 4 analog of A117330.

All terms are even. - Harvey P. Dale, May 05 2016

			a(1) = 880 =
  | 2  3  5  7|
  |11 13 17 19|
  |23 29 31 37|
  |41 43 47 53|.
a(10) = -2520 =
  |29 31 37 41|
  |43 47 53 59|
  |61 67 71 73|
  |79 83 89 97|.
a(21) = 0 =
  | 73  79  83  89|
  | 97 101 103 107|
  |109 113 127 131|
  |137 139 149 151|.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000040, A067276, A117301, A117330 , A118713.

A118799 := proc(n)
    local A,i,r,c ;
    A := Matrix(4,4) ;
    i := n ;
    for r from 1 to 4 do
    for c from 1 to 4 do
        A[r,c] := ithprime(i) ;
        i := i+1 ;
    end do:
    end do:
    LinearAlgebra[Determinant](A) ;
end proc: # R. J. Mathar, May 05 2013

Module[{nn=60,prs},prs=Prime[Range[nn]];Table[Det[Partition[ Take[ prs, {n,n+15}],4]],{n,nn-15}]] (* Harvey P. Dale, Apr 29 2016 *)

a(n) = matdet(matrix(4,4,i,j,prime((n+j-1)+4*(i-1)))); \\ Michel Marcus, Jan 25 2021

A118780 Semiprime(n)semiprime(n+3) - semiprime(n+1)semiprime(n+2), where semiprime(n) is the n-th semiprime.

Original entry on oeis.org

-14, -6, -5, 0, -7, -87, -4, 76, -8, -212, 64, -4, 128, 68, -265, 31, -12, -177, 104, 109, -28, 103, -101, -40, -24, -348, -176, 253, 81, -285, -97, 928, 364, -841, -257, -361, -127, -3, -125, 603, 359, -675, 367, -8, -860, 139, -3, 995, 280, -1276, -167, 629, 145, 443, -365, -579, 171, -569

Offset: 1

Jonathan Vos Post, May 22 2006

Semiprime analog of A117301.

By construction, every entry is also the difference between two 4-almost primes: a(1) = A014613(4)-A014613(5); a(2) = A014613(9)-A014613(11); a(3) = A014613(16)-A014613(18); a(4) = A014613(27)-A014613(27); etc. - R. J. Mathar, Nov 27 2007

			a(1) = -14 because the determinant of the first block of 4 consecutive semiprimes is:
|4. 6.|
|9. 10|.
a(4) = 0 because the determinant of the 4th block of 4 semiprimes is the first of a presumably infinite number of singular matrices:
|10. 14.|
|15. 21.|.
a(8) = 76, the first positive value in the sequence:
|22. 25.|
|26. 33.|.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A067276, A117301, A118713.

A001358 := proc(n) option remember ; local a; if n = 1 then 4 ; else for a from A001358(n-1)+1 do if numtheory[bigomega](a)= 2 then RETURN(a) ; fi ; od: fi ; end: A118780 := proc(n) A001358(n)*A001358(n+3)-A001358(n+1)*A001358(n+2) ; end: seq(A118780(n),n=1..58) ; # R. J. Mathar, Nov 27 2007

nmax = 58; spmax = nmax; SP = {};
While[nmax+3 > Length[SP], spmax += nmax; SP = Select[Range[spmax], PrimeOmega[#] == 2&]];
a[n_] := SP[[n]] SP[[n+3]] - SP[[n+1]] SP[[n+2]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Aug 01 2023 *)
#[[1]]#[[4]]-#[[2]]#[[3]]&/@Partition[Select[Range[300],PrimeOmega[#]==2&],4,1] (* Harvey P. Dale, Sep 08 2024 *)

a(n) = A001358(n)*A001358(n+3) - A001358(n+1)*A001358(n+2).

Better definition from Jens Kruse Andersen, May 03 2008

A118781 Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive semiprimes.

Original entry on oeis.org

-196, 272, -251, 149, -423, 909, -408, -452, 958, -123, -112, -460, 84, -271, -187, -162, 63, 7, 101, -483, -133, 205, -860, -46, 339, 1178, 848, 366, 1084, 719, -384, 334, -2736, -984, -1912, 214, 34, 40, -1735, -60, 64, -2263, -3468, 5795, 69, 132, 3007, 256, 2130, 3428

Offset: 1

Jonathan Vos Post, May 22 2006

Semiprime analog of A117330 Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive primes. The terminology "continuous" is used to distinguish from "discrete" which would be (in this 3 X 3 semiprime case) block 1: 4, 6, 9, 10, 14, 15, 21, 22, 25; block 2: 26, 33, 34, 35, 38, 39, 46, 49, 51; and so forth.

			a(1) = -196 because the determinant of the first continuous block of 9 semiprimes is:
| 4. 6. 9.|
|10. 14. 15.|
|21. 22. 25.|.
a(9) = 958 because the determinant of the 9th continuous block of 9 semiprimes is:
|25. 26. 33.|
|34. 35. 38.|
|39. 46. 49.|.
a(50) = 3428 because the determinant of the 50th continuous block of 9 semiprimes is:
|146. 155. 158.|
|159. 161. 166.|
|169. 177. 178.|.

Cf. A001358, A067276, A117301, A118713.

A118781 := proc(n)
    local A,i,r,c ;
    A := Matrix(3,3) ;
    i := n ;
    for r from 1 to 3 do
    for c from 1 to 3 do
        A[r,c] := A001358(i) ;
        i := i+1 ;
    end do:
    end do:
    LinearAlgebra[Determinant](A) ;
end proc: # R. J. Mathar, May 05 2013

Det/@(Partition[#,3]&/@(Partition[Select[Range[200],PrimeOmega[ #] == 2&],9,1])) (* Harvey P. Dale, Nov 29 2015 *)

a(n) = s(n)*s(n+4)*s(n+8) - s(n)*s(n+5)*s(n+7) - s(n+1)*s(n+3)*s(n+8) + s(n+1)*s(n+5)*s(n+6) + s(n+2)*s(n+3)*s(n+7) - s(n+2)*s(n+4)*s(n+6) where s(n) = A001358(n) is the n-th semiprime.

A118815 Determinants of 5 X 5 matrices consisting of 25 consecutive primes.

Original entry on oeis.org

-4656, 1440, 2912, 2832, -10464, -768, -17376, 20384, -72976, -18944, 12672, 41184, -199776, 28944, -21104, 3552, 25488, 338448, -60192, 39952, -21792, -161904, 499488, -83424, -7440, 7440, -54288, -75456, 1641792, 42288

Offset: 1

Jonathan Vos Post, May 23 2006

5 X 5 analog of A117330.

			a(1) = -4656 =
  | 2  3  5  7 11|
  |13 17 19 23 29|
  |31 37 41 43 47|
  |53 59 61 67 71|
  |73 79 83 89 97|.

Cf. A000040, A067276, A117301, A117330, A118713.

a(n) = matdet(matrix(5,5,i,j,prime((n+j-1)+5*(i-1)))); \\ Michel Marcus, Jan 25 2021

A118877 Determinant of n-th continuous block of 4 consecutive composites.

Original entry on oeis.org

-12, -12, 6, 6, -18, -18, 12, 12, -24, -24, 18, -3, -28, -2, -2, 24, 24, -36, -36, -2, -2, 32, -3, -42, 36, 36, -48, -48, 42, -3, -52, -2, -2, 48, -3, -58, -2, -2, 54, 54, -66, -66, -2, -2, 62, -3, -72, 66, 66, -78, -78, -2, -2, 74, -3, -84, 78, -3, -88, -2, -2

Offset: 1

Jonathan Vos Post, May 24 2006

Composites analog of A117301 Determinants of 2 X 2 matrices of continuous blocks of 4 consecutive primes. See also: A118780 Determinants of 2 X 2 matrices of continuous blocks of 4 consecutive semiprimes. The terminology "continuous" is used to distinguish from "discrete" which would be (in this composites case) block 1: 4, 6, 8, 9; block 2: 10, 12, 14, 15 and so forth. It is not until a(12) that we break the pattern of a(2n)=a(2n-1); there seem to be strangely many such pairs of two identical values. a(12) is also the first odd value in the sequence and the first prime.

			a(1) = -12 =
|4 6|
|8 9|.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002808, A117301, A118780.

Module[{nn=100,cmps},cmps=Select[Range[nn],CompositeQ];Det[ ArrayReshape[ #,{2,2}]]&/@Table[Take[cmps,{n,n+3}],{n,Length[cmps]-3}]] (* Harvey P. Dale, Aug 03 2020 *)

a(n) = A002808(n)*A002808(n+3) - A002808(n+1)*A002808(n+2).

A118983 Determinant of 3 X 3 matrices of n-th continuous block of 9 consecutive composites.

Original entry on oeis.org

24, 12, 0, 15, 30, 18, -4, -4, 34, -4, -4, 22, 8, 8, 0, -8, -8, 38, 4, 4, 26, 4, 4, 42, -4, -4, 58, -4, -4, 50, 4, 7, -7, -4, 52, 8, 8, 0, -8, -8, 68, 4, 4, 56, 4, 4, 80, -8, -8, 80, 4, 4, -4, 0, 4, -4, -4, 86, 4, 7

Offset: 1

Jonathan Vos Post, May 25 2006

Analog of A117330 with composites instead of primes.

			a(1) = 24 =
  | 4   6   8|
  | 9  10  12|
  |14  15  16|.
a(3) = 0 because of the first of an infinite number of singular matrices:
  | 8   9  10|
  |12  14  15|
  |16  18  20|.
a(15) = 0 because of the singular matrix:
  |25  26  27|
  |28  30  32|
  |33  34  35|.
a(38) = 0 because of the singular matrix:
  |55  56  57|
  |58  60  62|
  |63  64  65|.
a(54) = 0 because of the singular matrix:
  |76  77  78|
  |80  81  82|
  |84  85  86|.

Cf. A002808, A117301, A117330, A118780, A118781.

A118983 := proc(n) A002808(n)*A002808(n+4)*A002808(n+8) -A002808(n)*A002808(n+5) *A002808(n+7) -A002808(n+1)*A002808(n+3) *A002808(n+8) +A002808(n+1)*A002808(n+5) *A002808(n+6) +A002808(n+2)*A002808(n+3) *A002808(n+7) -A002808(n+2)*A002808(n+4) *A002808(n+6) ; end proc:
seq(A118983(n),n=1..60) ; # R. J. Mathar, Dec 22 2010

Det[#]&/@(Partition[#,3]&/@Partition[Select[Range[100],CompositeQ],9,1]) (* Harvey P. Dale, May 16 2019 *)

c(n) = for(k=0, primepi(n), isprime(n++)&&k--); n; \\ A002808
a(n) = matdet(matrix(3,3,i,j,c((n+j-1)+3*(i-1)))); \\ Michel Marcus, Jan 25 2021

a(n) = c(n)*c(n+4)*c(n+8) - c(n)*c(n+5)*c(n+7) - c(n+1)*c(n+3)*c(n+8) + c(n+1)*c(n+5)*c(n+6) + c(n+2)*c(n+3)*c(n+7) - c(n+2)*c(n+4)*c(n+6) where c(n) = A002808(n) is the n-th composite.

A370455 a(n) = greatest m such that 2^m divides prime(n+1)prime(n+2) - prime(n)prime(n+3).

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 3, 1, 4, 1, 3, 3, 3, 1, 2, 4, 1, 4, 1, 1, 2, 2, 1, 1, 3, 2, 3, 3, 3, 2, 1, 2, 3, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 4, 1, 3, 1, 3, 7, 1, 2, 2, 2, 3, 2, 4, 1, 3, 1, 2, 1, 3, 3, 3, 1, 2, 2, 1, 5, 2, 2, 1, 1, 2, 2, 1, 5, 1, 1, 3, 3, 2, 1, 2, 2, 1

Offset: 1

Clark Kimberling, Feb 26 2024

nonn

			prime(4)*prime(5) - prime(3)*prime(6) = 7*11 - 5*13 = 12, which is divisible by 2^2 but not 2^3, so a(3) = 2.

Cf. A007814, A117301.

p[n_] := Prime[n];
u = Table[p[n + 1] p[n + 2] - p[n] p[n + 3], {n, 1, 2000}];  (* A117302 *)
s[n_] := Last[Select[Range[15], IntegerQ[u[[n]]/2^#] &]];
Table[s[n], {n, 1, 200}]

a(n) = valuation(prime(n+1)*prime(n+2) - prime(n)*prime(n+3), 2); \\ Michel Marcus, Mar 01 2024

from sympy import prime
def A370455(n): return (~(m:=prime(n+1)*prime(n+2)-prime(n)*prime(n+3)) & m-1).bit_length() # Chai Wah Wu, Mar 02 2024

A118873 Determinant of n-th continuous block of 4 consecutive squares of primes.

Original entry on oeis.org

-29, -136, -1704, -6288, -5160, -14928, 52080, -97968, -84000, 98112, -524400, -84048, 637488, 231288, -1558440, -343200, 844152, -2799840, 1152360, 1469160, -783240, 4153800, -4254000, -11947320, -498768, -264360, -559248, 32952432, -2061360, -37128408, -10466400, 18355512

Offset: 1

Jonathan Vos Post, May 24 2006

Quadratic analog of A117301 Determinants of 2 X 2 matrices of continuous blocks of 4 consecutive primes. See also: A001248 Squares of primes. The terminology "continuous" is used to distinguish from "discrete" which would be block 1: 4, 9, 25, 49; block 2: 121, 169, 289, 361; and so forth. Through n = 10^6, the number of negative values a(n) in this sequence appears to be consistently larger than the number of positive values.

			a(1) = -29 =
  | 4   9|
  |25  49|.

Cf. A000040, A001248, A117301.

a:= n-> LinearAlgebra[Determinant](Matrix(2, (i,j)-> ithprime(n+2*i-3+j)^2)):
seq(a(n), n=1..32);  # Alois P. Heinz, Jan 25 2021

m = 32; p = Prime[Range[m + 3]]^2; Table[Det @ Partition[p[[n ;; n + 3]], 2], {n, 1, m}] (* Amiram Eldar, Jan 25 2021 *)

a(n) = prime(n)^2*prime(n+3)^2 - prime(n+1)^2*prime(n+2)^2; \\ Michel Marcus, Jan 25 2021

a(n) = prime(n)^2*prime(n+3)^2 - prime(n+1)^2*prime(n+2)^2.

A118875 Determinant of n-th continuous block of 9 consecutive squares of primes.

Original entry on oeis.org

-213720, 114432, -548352, 892800, -1774080, -7289856, 10105344, -79557120, -97790976, 171740160, 147556224, 56531520, -380053440, 122206464, -164292480, -958000320, 394761600, 189907200, 1139760000, -3023127360, -1495428480, -4260988800, -14501393280, 7022695680

Offset: 1

Jonathan Vos Post, May 24 2006

Quadratic analog of A117330 Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive primes. See also: A001248 Squares of primes. The terminology "continuous" is used to distinguish from "discrete" which would be block 1: 4, 9, 25, 49, 121, 169, 289, 361, 529; block 2: 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721; and so forth.

			a(1) = -213720 =
  |  4    9   25|
  | 49  121  169|
  |289  361  529|.
a(2) =
  |   9  25  49|
  | 121 169 289|
  | 361 529 841|.

Cf. A000040, A001248, A117301, A117330.

a:= n-> LinearAlgebra[Determinant](Matrix(3, (i,j)-> ithprime(n+3*i-4+j)^2)):
seq(a(n), n=1..25);  # Alois P. Heinz, Jan 25 2021

m = 24; p = Prime[Range[m + 8]]^2; Table[Det @ Partition[p[[n ;; n + 8]], 3], {n, 1, m}] (* Amiram Eldar, Jan 25 2021 *)

a(n) = matdet(matrix(3, 3, i, j, prime((n+j-1)+3*(i-1))^2)); \\ Michel Marcus, Jan 25 2021

Wrong Formula and data corrected by Michel Marcus, Jan 25 2021

A118876 Determinant of n-th continuous block of 16 consecutive squares of primes.

Original entry on oeis.org

768280320, -1010949120, -4719098880, -1791590400, 24298444800, -19462947840, -109685145600, -3192514560, 144441833472, -198529367040, 15778022400, 159125783040, 861983659008, 1193361776640, 1359501373440, 5328357672960

Offset: 1

Jonathan Vos Post, May 24 2006

Quadratic analog of A118799 Determinants of 4 X 4 matrices of continuous blocks of 16 consecutive primes. See also: A001248 Squares of primes. The terminology "continuous" is used to distinguish from "discrete" which would be block 1: 4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809; block 2: 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161; and so forth.

			a(1) = 768280320 =
|...4.....9...25....49.|
|.121...169..289...361.|
|.529...841..961..1369.|
|1681..1849.2209..2809.|.

Cf. A000040, A001248, A117301, A117330, A118799.

cp's OEIS Frontend

A118799 Determinants of 4 X 4 matrices of 16 consecutive primes.

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A118780 Semiprime(n)*semiprime(n+3) - semiprime(n+1)*semiprime(n+2), where semiprime(n) is the n-th semiprime.

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A118781 Determinants of 3 X 3 matrices of continuous blocks of 9 consecutive semiprimes.

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A118815 Determinants of 5 X 5 matrices consisting of 25 consecutive primes.

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A118877 Determinant of n-th continuous block of 4 consecutive composites.

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A118983 Determinant of 3 X 3 matrices of n-th continuous block of 9 consecutive composites.

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A370455 a(n) = greatest m such that 2^m divides prime(n+1)*prime(n+2) - prime(n)*prime(n+3).

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A118873 Determinant of n-th continuous block of 4 consecutive squares of primes.

A118780 Semiprime(n)semiprime(n+3) - semiprime(n+1)semiprime(n+2), where semiprime(n) is the n-th semiprime.

A370455 a(n) = greatest m such that 2^m divides prime(n+1)prime(n+2) - prime(n)prime(n+3).