A037165 -id:A037165 - cp's OEIS frontend

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

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

Artur Jasinski, Apr 05 2008

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

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

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

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 *)

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

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

Artur Jasinski, Apr 05 2008

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

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

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

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 *)

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

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

Artur Jasinski, Apr 05 2008

nonn

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

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

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

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 *)

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

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

Artur Jasinski, Apr 05 2008

nonn

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

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

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

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 *)

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

A069756 Frobenius number of the numerical semigroup generated by consecutive squares.

Original entry on oeis.org

23, 119, 359, 839, 1679, 3023, 5039, 7919, 11879, 17159, 24023, 32759, 43679, 57119, 73439, 93023, 116279, 143639, 175559, 212519, 255023, 303599, 358799, 421199, 491399, 570023, 657719, 755159, 863039, 982079, 1113023, 1256639, 1413719, 1585079, 1771559

Offset: 2

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 05 2002

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1, ..., a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive squares are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generated semigroup has the formula ab-a-b.

Given the set {n, n+1, n+2, n+3} and starting at n=0, the sum of all possible products of the terms in all possible subsets = a(n+2). Example for n=5, 5+6+7+8=26; 5(6+7+8)+6*(7+8)+7*8=277; 5*(6*7+6*8+7*8)+6*7*8=1066; 5*6*7*8=1680 and the sum of these 15 possible subsets is 3023 = a(5+2) = a(7). The sum is a(n+2) = n^4 + 10*n^3 + 35*n^2 + 50*n + 23. - J. M. Bergot, Apr 17 2013

			a(2)=23 because 23 is not a nonnegative linear combination of 4 and 9, but all integers greater than 23 are.

T. D. Noe, Table of n, a(n) for n = 2..1000
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000290, A037165, A059769, A069755, A069757-A069764.

seq(n^4+2*n^3-n^2-2*n-1, n=2..50); # Robert Israel, Nov 01 2015

Table[(n^2-1)((n+1)^2-1)-1, {n,2,30}] (* T. D. Noe, Nov 27 2006 *)
FrobeniusNumber/@Partition[Range[2,40]^2,2,1] (* Harvey P. Dale, Jul 25 2012 *)

x='x+O('x^50); Vec(x^2*(23+4*x-6*x^2+4*x^3-x^4)/(1-x)^5) \\ Altug Alkan, Nov 01 2015

a(n) = n^2*(n+1)^2-n^2-(n+1)^2 = n^4+2*n^3-n^2-2*n-1.
a(n) = Numerator of ((n + 2)! - (n - 2)!)/n!, n >=2. - Artur Jasinski, Jan 09 2007
G.f.: x^2*(23+4*x-6*x^2+4*x^3-x^4)/(1-x)^5. [Colin Barker, Feb 14 2012]
a(n) = (n-1)*n*(n+1)*(n+2) - 1 = A052762(n+2) - 1. - Jean-Christophe Hervé, Nov 01 2015

Corrected by T. D. Noe, Nov 27 2006

A096345 Primes of the form p*q - p - q, where p and q are two successive primes.

Original entry on oeis.org

7, 23, 59, 191, 839, 1439, 1931, 5039, 8447, 11447, 23399, 26891, 36479, 41579, 46619, 57119, 59999, 77279, 110879, 163199, 232307, 323759, 370871, 414731, 470579, 521267, 566999, 606791, 664199, 678971, 776159, 824459, 835379, 879839, 919631, 1183739, 1190279

Offset: 1

Giovanni Teofilatto, Jun 29 2004

nonn

These primes are not the sum of two squares.

The number of terms less than 10^n: 1, 3, 5, 9, 18, 35, 83, 190, 425, 1105, 2962, 7695, 20187, 54280, 147464, 402660, 1116912, ..., . - Robert G. Wilson v, Apr 09 2008

			a(3)=59 because 7*11 - 7 - 11=59.

Robert G. Wilson v, Table of n, a(n) for n=1..7700

Primes in A037165.

lst = {}; p = q = 2; Do[p = q; q = NextPrime@q; r = p*q - p - q; If[ PrimeQ@r, AppendTo[lst, r]], {n, 300}]; lst (* Robert G. Wilson v, Apr 09 2008 *)
Select[Times@@#-#[[1]]-#[[2]]&/@Partition[Prime[Range[300]],2,1],PrimeQ] (* Harvey P. Dale, Feb 17 2025 *)

More terms from Robert G. Wilson v, Jul 01 2004

A069757 Frobenius number of the numerical semigroup generated by three consecutive pentagonal numbers.

Original entry on oeis.org

43, 133, 287, 1699, 921, 1569, 3006, 3197, 4129, 12915, 6445, 8621, 14087, 13549, 16753, 43144, 20783, 25793, 38854, 35769, 43321, 101747, 48147, 57764, 82815, 74393, 89017, 198120, 93689, 108983, 151478, 133957, 159025, 341659, 162180

Offset: 2

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 05 2002

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the greatest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since three consecutive pentagonal numbers are relatively prime, they generate a numerical semigroup with a Frobenius number.

			a(2)=43 because 43 is not a nonnegative linear combination of 5, 12 and 22, but all integers greater than 43 are.

Harvey P. Dale, Table of n, a(n) for n = 2..1000
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Cf. A000326, A037165, A059769, A069755-A069764.

FrobeniusNumber/@Partition[PolygonalNumber[5,Range[2,40]],3,1] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 16 2018 *)

A069759 Frobenius number of the numerical semigroup generated by consecutive hex numbers.

Original entry on oeis.org

107, 647, 2159, 5399, 11339, 21167, 36287, 58319, 89099, 130679, 185327, 255527, 343979, 453599, 587519, 749087, 941867, 1169639, 1436399, 1746359, 2103947, 2513807, 2980799, 3509999, 4106699, 4776407

Offset: 1

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 08 2002

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive hex numbers are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generated semigroup has the formula ab-a-b.

			a(1)=107 because 107 is not a nonnegative linear combination of 7 and 19, but all integers greater than 107 are.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A003215, A037165, A059769, A069755-A069764.

FrobeniusNumber/@Partition[Table[3n^2+3n+1,{n,30}],2,1] (* Harvey P. Dale, Dec 25 2018 *)

a(n) = 9*n^4+36*n^3+45*n^2+18*n-1; with offset 2, a(n) = 9*n^4-9*n^2-1.

G.f.: x*(107+112*x-6*x^2+4*x^3-x^4)/(1-x)^5. - Colin Barker, Feb 14 2012

A069761 Frobenius number of the numerical semigroup generated by four consecutive tetrahedral numbers.

Original entry on oeis.org

41, 249, 253, 853, 1243, 1571, 2619, 5059, 5357, 9437, 11801, 13609, 18327, 27607, 28919, 41951, 49169, 54473, 67253, 90573, 94051, 124099, 140347, 152027, 178989, 226141, 233369, 291089, 321839, 343639, 392631, 475999, 488993, 587633, 639653, 676181, 756779

Offset: 2

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 09 2002

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since four consecutive tetrahedral numbers are relatively prime, they generate a numerical semigroup with a Frobenius number.

			a(2) = 41 because 41 is not a nonnegative linear combination of 4, 10, 20 and 35, but all integers greater than 43 are.

Harvey P. Dale, Table of n, a(n) for n = 2..100
R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Cf. A000292, A037165, A059769, A069755-A069764.

FrobeniusNumber/@Partition[Binomial[Range[2,50]+2,3],4,1] (* Harvey P. Dale, Jan 22 2012 *)

Conjecture: a(n)= +a(n-1) +4*a(n-6) -4*a(n-7) -6*a(n-12) +6*a(n-13) +4*a(n-18) -4*a(n-19) -a(n-24) +a(n-25). - R. J. Mathar, Aug 15 2025

Conjectured g.f.: x^2*(-4*x^2 -600*x^3 -390*x^4 -1680*x^9 -282*x^8 -496*x^11 -804*x^10 -208*x -312*x^15 -144*x^14 -768*x^13 -772*x^12-41 -32*x^18 -40*x^17 -102*x^16 -2*x^20 -8*x^19 -1608*x^7 +x^24 -884*x^6 -328*x^5) / ( (1+x)^4 *(x^2-x+1)^4 *(1+x+x^2)^4 *(x-1)^5 ). - R. J. Mathar, Aug 15 2025

Sequence terms corrected and extended by Harvey P. Dale, Jan 22 2012

Offset corrected and example corrected by Harvey P. Dale, Jan 24 2012

A069762 Frobenius number of the numerical semigroup generated by three consecutive pyramidal numbers.

Original entry on oeis.org

51, 191, 609, 1324, 2813, 4711, 8576, 13894, 23319, 34165, 51661, 71126, 100529, 136239, 187543, 241586, 321251, 404839, 516704, 645358, 813141, 982651, 1221299, 1463734, 1767473, 2106271, 2524101, 2940909, 3500209, 4061663, 4736456, 5474526, 6352219, 7228469

Offset: 2

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 18 2002

Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since three consecutive pyramidal numbers are relatively prime, they generate a numerical semigroup with a Frobenius number.

			a(2)=51 because 51 is not a nonnegative linear combination of 5, 14 and 30, but all integers greater than 51 are.

R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Cf. A000330, A037165, A059769, A069755-A069764.

More terms from and offset corrected by Sean A. Irvine, May 19 2024

cp's OEIS Frontend

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

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

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

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

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A069756 Frobenius number of the numerical semigroup generated by consecutive squares.

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A096345 Primes of the form p*q - p - q, where p and q are two successive primes.

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A069757 Frobenius number of the numerical semigroup generated by three consecutive pentagonal numbers.

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A069759 Frobenius number of the numerical semigroup generated by consecutive hex numbers.