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.

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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 *)

A138984 a(n) = Frobenius number for 4 successive numbers = F(n+1, n+2, n+3, n+4).

Original entry on oeis.org

1, 2, 3, 9, 11, 13, 23, 26, 29, 43, 47, 51, 69, 74, 79, 101, 107, 113, 139, 146, 153, 183, 191, 199, 233, 242, 251, 289, 299, 309, 351, 362, 373, 419, 431, 443, 493, 506, 519, 573, 587, 601, 659, 674, 689, 751, 767, 783, 849, 866, 883, 953, 971, 989, 1063, 1082
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Examples

			a(4) = 9 because 9 is the largest number k such that the equation 5*x_1 + 6*x_2 + 7*x_3 + 9*x_4 = k has no solution for any nonnegative x_i (in other words, for every k > 9 there exist one or more solutions).

Links

Crossrefs

Frobenius number for k successive numbers: A028387 (k=2), A079326 (k=3), this sequence (k=4), A138985 (k=5), A138986 (k=6), A138987 (k=7), A138988 (k=8).

Programs

  • Mathematica
    Table[FrobeniusNumber[{n + 1, n + 2, n + 3, n + 4}], {n, 1, 100}]
Table[n + Floor[(n-1)/3]*(n+1), {n,56}] (* Giorgos Kalogeropoulos, Apr 06 2025 *)

Formula

G.f.: x*(x^6-4*x^3-x^2-x-1) / ((x-1)^3*(x^2+x+1)^2). [Colin Barker, Dec 13 2012]
a(n) = n + (n+1)*floor((n-1)/3). - Giorgos Kalogeropoulos, Apr 06 2025

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

A204557 Right edge of the triangle A045975.

Original entry on oeis.org

1, 4, 21, 36, 85, 120, 217, 280, 441, 540, 781, 924, 1261, 1456, 1905, 2160, 2737, 3060, 3781, 4180, 5061, 5544, 6601, 7176, 8425, 9100, 10557, 11340, 13021, 13920, 15841, 16864, 19041, 20196, 22645, 23940, 26677, 28120, 31161, 32760, 36121, 37884, 41581
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 18 2012

Keywords

Links

Programs

  • Haskell
    a204557 = last . a045975_row
  • Magma
    [n*(2*n^2+(3-(-1)^n)*n-(-1)^n-3)/4: n in [1..50]]; // G. C. Greubel, Jun 15 2018
  • Mathematica
    Table[n*(2*n^2+(3-(-1)^n)*n-(-1)^n-3)/4, {n, 1, 50}] (* G. C. Greubel, Jun 15 2018 *)
LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,4,21,36,85,120,217},50] (* Harvey P. Dale, Feb 20 2021 *)
  • PARI
    Vec(-x*(-1-3*x-14*x^2-6*x^3-x^4+x^5)/((1+x)^3*(x-1)^4) + O(x^100)) \\ Colin Barker, Jan 28 2016

Formula

a(n) = A045975(n,n);
a(n) = A079326(n+1) * n;
a(n) = A204556(n) + A045895(n).
G.f.: -x*(-1-3*x-14*x^2-6*x^3-x^4+x^5) / ((1+x)^3*(x-1)^4). - R. J. Mathar, Aug 13 2012
From Colin Barker, Jan 28 2016: (Start)
a(n) = n*(2*n^2+(3-(-1)^n)*n-(-1)^n-3)/4.
a(n) = (n^3+n^2-2*n)/2 for n even.
a(n) = (n^3+2*n^2-n)/2 for n odd.
(End)

A386243 a(n) is the smallest possible g(k) in a set of increasing numbers g(1) < g(2) < ... < g(k) having Frobenius number n.

Original entry on oeis.org

3, 5, 5, 7, 4, 7, 5, 9, 7, 9, 5, 9, 8, 11, 10, 11, 7, 13, 6, 9, 11, 10, 7, 13, 11, 11, 8, 13, 7, 13, 9, 13, 14, 13, 11, 13, 12, 13, 11, 17, 8, 17, 12, 17, 14, 16, 9, 19, 11, 17, 14, 17, 10, 13, 9, 17, 15, 17, 11, 18, 15, 19, 16, 15, 12, 16, 16, 18, 11, 17, 10, 19, 17, 17, 18, 18, 15
Offset: 1

Views

Author

Gordon Hamilton, Jul 16 2025

Keywords

Examples

			a(15) = 10 because the set {6,7,10} has the Frobenius number of 15. No set of the form {..., 9} or {..., 8}, etc. has a Frobenius number of 15.

Links

Crossrefs

Cf. A028387, A079326, A124506, A138984, A138985.

Extensions

More terms from David A. Corneth, Jul 16 2025

A151898 First differences of Frobenius numbers for 7 successive numbers A138987.

Original entry on oeis.org

1, 1, 1, 1, 1, 9, 2, 2, 2, 2, 2, 16, 3, 3, 3, 3, 3, 23, 4, 4, 4, 4, 4, 30, 5, 5, 5, 5, 5, 37, 6, 6, 6, 6, 6, 44, 7, 7, 7, 7, 7, 51, 8, 8, 8, 8, 8, 58, 9, 9, 9, 9, 9, 65, 10, 10, 10, 10, 10, 72, 11, 11, 11, 11, 11, 79, 12, 12, 12, 12, 12, 86, 13, 13, 13, 13, 13, 93, 14, 14, 14, 14, 14, 100, 15
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Comments

First differences of Frobenius numbers for 2 successive numbers see A005843
First differences of Frobenius numbers for 3 successive numbers see A014682
First differences of Frobenius numbers for 4 successive numbers see A138995
First differences of Frobenius numbers for 5 successive numbers see A138996
First differences of Frobenius numbers for 6 successive numbers see A138997
First differences of Frobenius numbers for 7 successive numbers see A151898
First differences of Frobenius numbers for 8 successive numbers see A138999

Links

Crossrefs

Cf. A028387, A037165, A079326, A138985, A138986, A138987, A138988.
Cf. A138989, A138990, A138991, A138992, A138993, A138994.
Cf. A005843, A014682, A138995, A138996, A138997, A138999.

Programs

  • Mathematica
    a = {}; Do[AppendTo[a, FrobeniusNumber[{n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7}]], {n, 1, 100}]; Differences[a]
Differences[Table[FrobeniusNumber[Range[n,n+6]],{n,2,90}]] (* or *) LinearRecurrence[ {0,0,0,0,0,2,0,0,0,0,0,-1},{1,1,1,1,1,9,2,2,2,2,2,16},90] (* Harvey P. Dale, Jul 26 2024 *)

Formula

a(n) = A138987(n+1)-A138987(n).
G.f.: -x*(2*x^11-9*x^5-x^4-x^3-x^2-x-1) / ((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2). [Colin Barker, Dec 13 2012]

A296307 Array read by upwards antidiagonals: f(n,k) = (n+1)*ceiling(n/(k-1)) - 1.

Original entry on oeis.org

1, 5, 1, 11, 2, 1, 19, 7, 2, 1, 29, 9, 3, 2, 1, 41, 17, 9, 3, 2, 1, 55, 20, 11, 4, 3, 2, 1, 71, 31, 13, 11, 4, 3, 2, 1, 89, 35, 23, 13, 5, 4, 3, 2, 1, 109, 49, 26, 15, 13, 5, 4, 3, 2, 1, 131, 54, 29, 17, 15, 6, 5, 4, 3, 2, 1, 155, 71, 43, 29, 17, 15, 6, 5, 4, 3
Offset: 1

Views

Author

Gerhard Kirchner, Dec 10 2017

Keywords

Comments

f(n,k) = (n+1)*ceiling(n/(k-1))-1 is the Frobenius number F(n+1,n+2,...,n+k), k>1. This formula is derived in "Frobenius number for a set of successive numbers".
f(n,k) is the greatest number which is not a linear combination of n+1,n+2,...,n+k with nonnegative coefficients.
Example: f(2,3) = 5 because 6=2*3, 7=3+4, 8=2*4, 9=3*3, 10=2*3+4 and so on.
Special sequences: f(n,2) = A028387(n), f(n,3) = A079326(n+1), f(n,4) = A138984(n), f(n,5) = A138985(n), f(n,6) = A138986(n), f(n,7) = A138987(n), f(n,8) = A138988(n).
f(n,k) is a generalization of these sequences.

Examples

			Example:
   f(n,2)   f(n,3)   f(n,4)
  a(1)= 1   a(3)=1   a(6) =1
  a(2)= 5   a(5)=2   a(9) =2
  a(4)=11   a(8)=7   a(13)=3
More terms in "Table of Frobenius numbers".

Links

Crossrefs

Cf. A028387, A079326, A138984, A138985, A138986, A138987, A138988.
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