cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A037165 a(n) = prime(n)*prime(n+1) - prime(n) - prime(n+1).

Original entry on oeis.org

1, 7, 23, 59, 119, 191, 287, 395, 615, 839, 1079, 1439, 1679, 1931, 2391, 3015, 3479, 3959, 4619, 5039, 5615, 6395, 7215, 8447, 9599, 10199, 10811, 11447, 12095, 14111, 16379, 17679, 18767, 20423, 22199, 23399, 25271, 26891, 28551, 30615, 32039
Offset: 1

Views

Author

Armand Turpel (armandt(AT)unforgettable.com)

Keywords

Comments

a(n) is also the Frobenius number of the numerical semigroup generated by prime(n) and prime(n+1). - Victoria A Sapko (vsapko(AT)math.unl.edu), Feb 21 2001

Links

Crossrefs

Frobenius numbers for k successive primes: this sequence (k=2), A138989 (k=3), A138990 (k=4), A138991 (k=5), A138992 (k=6), A138993 (k=7), A138994 (k=8).
Cf. A001043, A006094.

Programs

  • Magma
    [NthPrime(n)*NthPrime(n+1)-NthPrime(n)-NthPrime(n+1): n in [1..45]]; // Vincenzo Librandi, Dec 18 2012
  • Mathematica
    f[n_] := FrobeniusNumber[{Prime[n], Prime[n + 1]}]; Array[f, 41] (* Robert G. Wilson v, Aug 04 2012 *)
Times@@#-Total[#]&/@Partition[Prime[Range[50]],2,1] (* Harvey P. Dale, Dec 27 2015 *)
  • PARI
    a(n)=my(p=prime(n),q=nextprime(p+1)); p*q-p-q \\ Charles R Greathouse IV, Apr 28 2015

Formula

a(n) = A006094(n) - A001043(n). - Michel Marcus, Mar 02 2019

A138990 a(n) = Frobenius number for 4 successive primes = F[p(n), p(n+1), p(n+2), p(n+3)].

Original entry on oeis.org

1, 4, 9, 23, 42, 67, 83, 101, 125, 199, 262, 335, 367, 393, 492, 593, 704, 807, 873, 990, 817, 950, 1101, 1353, 2039, 2624, 2371, 1494, 1431, 1640, 2927, 2368, 2875, 2667, 3570, 3348, 3625, 3918, 4531, 3816, 4831, 4543, 9357, 4819, 4131, 6611, 5735, 10483
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Examples

			a(3)=23 because 23 is the largest number k such that the equation 7*x_1 + 11*x_2 + 13*x_3 + 17*x + 4 = k has no solution for any nonnegative x_i (in other words, for every k > 23 there exist one or more solutions).

Links

Crossrefs

Frobenius numbers for k successive primes: A037165 (k=2), A138989 (k=3), this sequence (k=4), A138991 (k=5), A138992 (k=6), A138993 (k=7), A138994 (k=8).
Cf. A028387, A079326, A138985, A138986, A138987, A138988.

Programs

  • Mathematica
    Table[FrobeniusNumber[{Prime[n],Prime[n + 1], Prime[n + 2], Prime[n + 3]}], {n, 1, 100}]
FrobeniusNumber/@Partition[Prime[Range[60]],4,1] (* Harvey P. Dale, Nov 23 2014 *)

Extensions

Definition corrected by Harvey P. Dale, Aug 15 2014

A138991 a(n) = Frobenius number for 5 successive primes = F[p(n), p(n+1), p(n+2), p(n+3), p(n+4)].

Original entry on oeis.org

1, 4, 9, 23, 31, 54, 66, 101, 125, 143, 200, 261, 285, 307, 398, 434, 588, 563, 672, 708, 659, 717, 935, 1078, 1748, 1816, 1135, 1173, 1104, 1277, 1911, 1975, 2188, 2111, 2680, 2593, 2683, 3266, 2861, 3297, 3757, 3996, 4198, 3275, 2953, 3457, 4668, 6688
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Examples

			a(3)=23 because 23 is the largest number k such that the equation 7*x_1 + 11*x_2 + 13*x_3 + 17*x_4 + 19*x_5 = k has no solution for any nonnegative x_i (in other words, for every k > 23 there exist one or more solutions).

Crossrefs

Frobenius numbers for k successive primes: A037165 (k=2), A138989 (k=3), A138990 (k=4), this sequence (k=5), A138992 (k=6), A138993 (k=7), A138994 (k=8).
Cf. A028387, A079326, A138985, A138986, A138987, A138988.

Programs

  • Mathematica
    Table[FrobeniusNumber[{Prime[n],Prime[n + 1], Prime[n + 2], Prime[n + 3], Prime[n + 4]}], {n, 1, 100}]
FrobeniusNumber/@Partition[Prime[Range[80]],5,1] (* Harvey P. Dale, Aug 15 2014 *)

A138992 a(n) = Frobenius number for 6 successive primes = F[p(n), p(n+1), p(n+2), p(n+3), p(n+4), p(n+5)].

Original entry on oeis.org

1, 4, 9, 16, 31, 41, 64, 63, 102, 143, 169, 216, 203, 264, 304, 381, 470, 502, 538, 562, 592, 638, 769, 989, 1360, 1008, 929, 961, 995, 1051, 1530, 1582, 1777, 1694, 2084, 2140, 2369, 2288, 2527, 2778, 3399, 2721, 2859, 2698, 2756, 3035, 3613, 5800, 4765
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Examples

			a(4)=16 because 16 is the largest number k such that equation 7*x_1 + 11*x_2 + 13*x_3 + 17*x_4 + 19*x_5 + 23*x_6 = k has no solution for any nonnegative x_i (in other words, for every k > 16 there exist one or more solutions).

Crossrefs

Frobenius numbers for k successive primes: A037165 (k=2), A138989 (k=3), A138990 (k=4), A138991 (k=5), this sequence (k=6), A138993 (k=7), A138994 (k=8).
Cf. A028387, A079326, A138985, A138986, A138987, A138988.

Programs

  • Mathematica
    Table[FrobeniusNumber[{Prime[n],Prime[n + 1], Prime[n + 2], Prime[n + 3], Prime[n + 4], Prime[n + 5]}], {n, 1, 100}]
FrobeniusNumber/@Partition[Prime[Range[100]],6,1] (* Harvey P. Dale, Aug 15 2014 *)

A138993 a(n) = Frobenius number for 7 successive primes = F[p(n), p(n+1), p(n+2), p(n+3), p(n+4), p(n+5), p(n+6)].

Original entry on oeis.org

1, 4, 9, 16, 27, 41, 49, 63, 102, 114, 169, 187, 203, 221, 304, 328, 409, 441, 465, 495, 525, 559, 769, 811, 867, 907, 826, 854, 886, 938, 1403, 1451, 1505, 1555, 1786, 1838, 1741, 2125, 2193, 2605, 2325, 2005, 2479, 2318, 2362, 2637, 3402, 4012, 3857, 3666
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Examples

			a(4)=16 because 16 is the largest number k such that the equation 7*x_1 + 11*x_2 + 13*x_3 + 17*x_4 + 19*x_5 + 23*x_6 + 29*x_7 = k has no solution for any nonnegative x_i (in other words, for every k > 16 there exist one or more solutions).

Crossrefs

Frobenius numbers for k successive primes: A037165 (k=2), A138989 (k=3), A138990 (k=4), A138991 (k=5), A138992 (k=6), this sequence (k=7), A138994 (k=8).
Cf. A028387, A079326, A138985, A138986, A138987, A138988.

Programs

  • Mathematica
    Table[FrobeniusNumber[{Prime[n],Prime[n + 1], Prime[n + 2], Prime[n + 3], Prime[n + 4], Prime[n + 5], Prime[n + 6]}], {n, 1, 100}]
FrobeniusNumber/@Partition[Prime[Range[100]],7,1] (* Harvey P. Dale, Aug 15 2014 *)

A138994 a(n) = Frobenius number for 8 successive primes = F[p(n), p(n+1), p(n+2), p(n+3), p(n+4), p(n+5), p(n+6), p(n+7)].

Original entry on oeis.org

1, 4, 9, 16, 27, 35, 49, 63, 102, 114, 138, 150, 162, 221, 257, 275, 352, 368, 398, 424, 452, 559, 686, 633, 772, 705, 723, 747, 777, 938, 1149, 1189, 1231, 1406, 1637, 1536, 1741, 1799, 2193, 1913, 1967, 1824, 2099, 2125, 2165, 2438, 2769, 3347, 3403, 3212
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Examples

			a(4)=16 because 16 is the largest number k such that the equation 7*x_1 + 11*x_2 + 13*x_3 + 17*x_4 + 19*x_5 + 23*x_6 + 29*x_7 + 31*x_8 = k has no solution for any nonnegative x_i (in other words, for every k > 16 there exist one or more solutions).

Crossrefs

Frobenius numbers for k successive primes: A037165 (k=2), A138989 (k=3), A138990 (k=4), A138991 (k=5), A138992 (k=6), A138993 (k=7), this sequence (k=8).
Cf. A028387, A079326, A138985, A138986, A138987, A138988.

Programs

  • Mathematica
    Table[FrobeniusNumber[{Prime[n],Prime[n + 1], Prime[n + 2], Prime[n + 3], Prime[n + 4], Prime[n + 5], Prime[n + 6], Prime[n + 7]}], {n, 1, 100}]
FrobeniusNumber/@Partition[Prime[Range[100]],8,1] (* Harvey P. Dale, Aug 15 2014 *)

A138995 First differences of Frobenius numbers for 4 successive numbers A138984.

Original entry on oeis.org

1, 1, 6, 2, 2, 10, 3, 3, 14, 4, 4, 18, 5, 5, 22, 6, 6, 26, 7, 7, 30, 8, 8, 34, 9, 9, 38, 10, 10, 42, 11, 11, 46, 12, 12, 50, 13, 13, 54, 14, 14, 58, 15, 15, 62, 16, 16, 66, 17, 17, 70, 18, 18, 74, 19, 19, 78, 20, 20, 82, 21, 21, 86, 22, 22, 90, 23, 23, 94, 24, 24, 98, 25, 25, 102, 26
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Comments

For first differences of Frobenius numbers for 2 successive numbers see A005843
For first differences of Frobenius numbers for 3 successive numbers see A014682
For first differences of Frobenius numbers for 4 successive numbers see A138995
For first differences of Frobenius numbers for 5 successive numbers see A138996
For first differences of Frobenius numbers for 6 successive numbers see A138997
For first differences of Frobenius numbers for 7 successive numbers see A138998
For first differences of Frobenius numbers for 8 successive numbers see A138999

Links

Crossrefs

Cf. A028387, A037165, A079326, A138985, A138986, A138987, A138988, A138989, A138990, A138991, A138992, A138993, A138994, A138995, A138996, A138997, A138998, A138999.

Programs

  • Mathematica
    a = {}; Do[AppendTo[a, FrobeniusNumber[{n + 1, n + 2, n + 3, n + 4}]], {n, 1, 100}]; Differences[a]
LinearRecurrence[{0, 0, 2, 0, 0, -1}, {1, 1, 6, 2, 2, 10},50] (* G. C. Greubel, Feb 18 2017 *)
Differences[Table[FrobeniusNumber[Range[n,n+3]],{n,2,100}]] (* Harvey P. Dale, Dec 22 2018 *)
  • PARI
    x='x+O('x^50); Vec(-x*(2*x^5-6*x^2-x-1) / ((x-1)^2*(x^2+x+1)^2)) \\ G. C. Greubel, Feb 18 2017

Formula

a(n) = A138984(n+1) - A138984(n).
a(n) = 2*a(n-3) - a(n-6). - R. J. Mathar, Apr 20 2008
a(n) = (1/3)*x(mod(n,3))*mod(n,3)-(1/3)*n*x(mod(n,3))+(1/3)*n*x(3+mod(n,3))+x(mod(n,3))-(1/3)*mod(n,3)*x(3+mod(n,3)). - Alexander R. Povolotsky, Apr 20 2008
G.f.: -x*(2*x^5-6*x^2-x-1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, Dec 13 2012

A138996 First differences of Frobenius numbers for 5 successive numbers A138985.

Original entry on oeis.org

1, 1, 1, 7, 2, 2, 2, 12, 3, 3, 3, 17, 4, 4, 4, 22, 5, 5, 5, 27, 6, 6, 6, 32, 7, 7, 7, 37, 8, 8, 8, 42, 9, 9, 9, 47, 10, 10, 10, 52, 11, 11, 11, 57, 12, 12, 12, 62, 13, 13, 13, 67, 14, 14, 14, 72, 15, 15, 15, 77, 16, 16, 16, 82, 17, 17, 17, 87, 18, 18, 18, 92, 19, 19, 19, 97, 20, 20, 20
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Comments

For first differences of Frobenius numbers for 2 successive numbers see A005843
For first differences of Frobenius numbers for 3 successive numbers see A014682
For first differences of Frobenius numbers for 4 successive numbers see A138995
For first differences of Frobenius numbers for 5 successive numbers see A138996
For first differences of Frobenius numbers for 6 successive numbers see A138997
For first differences of Frobenius numbers for 7 successive numbers see A138998
For first differences of Frobenius numbers for 8 successive numbers see A138999

Links

Crossrefs

Cf. A028387, A037165, A079326, A138985, A138986, A138987, A138988, A138989, A138990, A138991, A138992, A138993, A138994, A138995, A138996, A138997, A138998, A138999.

Programs

  • Mathematica
    a = {}; Do[AppendTo[a, FrobeniusNumber[{n + 1, n + 2, n + 3, n + 4, n + 5}]], {n, 1, 100}]; Differences[a]
LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, -1}, {1, 1, 1, 7, 2, 2, 2,
  12}, 50] (* G. C. Greubel, Feb 18 2017 *)
  • PARI
    x='x+O('x^50); Vec(-x*(2*x^7-7*x^3-x^2-x-1) / ((x-1)^2*(x+1)^2*(x^2+1)^2)) \\ G. C. Greubel, Feb 18 2017

Formula

a(n) = A138985(n+1) - A138985(n).
a(n) = 2*a(n-4) - a(n-8). - R. J. Mathar, Apr 20 2008
a(n) = -(1/4)*mod(n,4)*x(4+mod(n,4))+(1/4)*n*x(4+mod(n,4))+x(mod(n,4))-(1/4)*n*x(mod(n,4))+(1/4)*mod(n,4)*x(mod(n,4)). - Alexander R. Povolotsky, Apr 20 2008
G.f.: -x*(2*x^7-7*x^3-x^2-x-1) / ((x-1)^2*(x+1)^2*(x^2+1)^2). - Colin Barker, Dec 13 2012

A138997 First differences of Frobenius numbers for 6 successive numbers A138986.

Original entry on oeis.org

1, 1, 1, 1, 8, 2, 2, 2, 2, 14, 3, 3, 3, 3, 20, 4, 4, 4, 4, 26, 5, 5, 5, 5, 32, 6, 6, 6, 6, 38, 7, 7, 7, 7, 44, 8, 8, 8, 8, 50, 9, 9, 9, 9, 56, 10, 10, 10, 10, 62, 11, 11, 11, 11, 68, 12, 12, 12, 12, 74, 13, 13, 13, 13, 80, 14, 14, 14, 14, 86, 15, 15, 15, 15, 92, 16, 16, 16, 16, 98, 17, 17
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Comments

For first differences of Frobenius numbers for 2 successive numbers see A005843
For first differences of Frobenius numbers for 3 successive numbers see A014682
For first differences of Frobenius numbers for 4 successive numbers see A138995
For first differences of Frobenius numbers for 5 successive numbers see A138996
For first differences of Frobenius numbers for 6 successive numbers see A138997
For first differences of Frobenius numbers for 7 successive numbers see A138998
For first differences of Frobenius numbers for 8 successive numbers see A138999

Links

Crossrefs

Cf. A028387, A037165, A079326, A138985, A138986, A138987, A138988, A138989, A138990, A138991, A138992, A138993, A138994, A138995, A138996, A138997, A138998, A138999.

Programs

  • Mathematica
    a = {}; Do[AppendTo[a, FrobeniusNumber[{n + 1, n + 2, n + 3, n + 4, n + 5, n + 6}]], {n, 1, 100}]; Differences[a]
LinearRecurrence[{0, 0, 0, 0, 2, 0, 0, 0, 0, -1}, {1, 1, 1, 1, 8, 2,
  2, 2, 2, 14}, 50] (* G. C. Greubel, Feb 18 2017 *)
Differences[Table[FrobeniusNumber[Range[n,n+5]],{n,2,90}]] (* Harvey P. Dale, Dec 18 2023 *)
  • PARI
    x='x + O('x^50); Vec(-(-1-x-x^2-x^3-8*x^4+2*x^9)/((x-1)^2*(x^4+x^3+x^2+x+1)^2)) \\ G. C. Greubel, Feb 18 2017

Formula

a(n) = A138986(n+1) - A138986(n).
O.g.f.= -(-1-x-x^2-x^3-8*x^4+2*x^9)/((x-1)^2*(x^4+x^3+x^2+x+1)^2). - R. J. Mathar, Apr 20 2008
a(n) = 2*a(n-5) - a(n-10). - R. J. Mathar, Apr 20 2008
a(n)= (1/5)*n*x(5+mod(n,5))-(1/5)*mod(n,5)*x(5+mod(n,5))+x(mod(n,5))-(1/5)*n*x(mod(n,5))+(1/5) *mod(n,5)*x(mod(n,5)). - Alexander R. Povolotsky, Apr 20 2008

A138999 First differences of Frobenius numbers for 8 successive numbers A138988.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 10, 2, 2, 2, 2, 2, 2, 18, 3, 3, 3, 3, 3, 3, 26, 4, 4, 4, 4, 4, 4, 34, 5, 5, 5, 5, 5, 5, 42, 6, 6, 6, 6, 6, 6, 50, 7, 7, 7, 7, 7, 7, 58, 8, 8, 8, 8, 8, 8, 66, 9, 9, 9, 9, 9, 9, 74, 10, 10, 10, 10, 10, 10, 82, 11, 11, 11, 11, 11, 11, 90, 12, 12, 12, 12, 12, 12, 98, 13, 13, 13, 13
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Comments

For first differences of Frobenius numbers for 2 successive numbers see A005843
For first differences of Frobenius numbers for 3 successive numbers see A014682
For first differences of Frobenius numbers for 4 successive numbers see A138995
For first differences of Frobenius numbers for 5 successive numbers see A138996
For first differences of Frobenius numbers for 6 successive numbers see A138997
For first differences of Frobenius numbers for 7 successive numbers see A138998
For first differences of Frobenius numbers for 8 successive numbers see A138999

Links

Crossrefs

Cf. A028387, A037165, A079326, A138985, A138986, A138987, A138988, A138989, A138990, A138991, A138992, A138993, A138994, A138995, A138996, A138997, A138998, A138999.

Programs

  • Mathematica
    a = {}; Do[AppendTo[a, FrobeniusNumber[{n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8}]], {n, 1, 100}]; Differences[a]
Differences[Table[FrobeniusNumber[Range[n,n+7]],{n,2,90}]] (* Harvey P. Dale, Oct 02 2011 *)

Formula

a(n) = A138988(n+1) - A138988(n).
From R. J. Mathar, Apr 20 2008: (Start)
G.f.: -(-1-x-x^2-x^3-x^4-x^5-10*x^6+2*x^13)/((x-1)^2*(x^6+x^5+x^4+x^3+x^2+x+1)^2).
a(n) = 2*a(n-7) - a(n-14).
(End)
a(n) = -(1/7)*mod(n,7)*x(7+mod(n,7))+(1/7)*mod(n,7)*x(mod(n,7))+x(mod(n,7))-(1/7)*n *x(mod(n,7))+(1/7)*n*x(7+mod(n,7)). - Alexander R. Povolotsky, Apr 20 2008
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