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 27 results. Next

A137367 Subset of A037165 (p(n)*p(n+1)-p(n)-p(n+1)) for twin primes.

Original entry on oeis.org

7, 23, 119, 287, 839, 1679, 3479, 5039, 10199, 11447, 18767, 22199, 32039, 36479, 38807, 51527, 57119, 72359, 78959, 96719, 120407, 175559, 185759, 212519, 271439, 323759, 358799, 380687, 410879, 434279, 654479, 674039, 683927, 734447, 776159
Offset: 1

Views

Author

Zak Seidov, Apr 09 2008

Keywords

Examples

			3*5-3-5=7, 5*7-5-7=23, 11*13-11-13=119.

Links

Crossrefs

Cf. A037165.

Programs

  • Mathematica
    ss={7};Do[If[PrimeQ[p1=6m-1]&&PrimeQ[p2=6m+1],p=-1-12 m+36 m^2;AppendTo[ss,p]],{m,300}];ss
Times@@#-Total[#]&/@Select[Partition[Prime[Range[200]],2,1],#[[2]]-#[[1]] == 2&] (* Harvey P. Dale, Jun 14 2014 *)

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

A138989 a(n) = Frobenius number for 3 successive primes = F[p(n), p(n+1), p(n+2)].

Original entry on oeis.org

1, 4, 13, 30, 53, 80, 117, 131, 194, 286, 293, 520, 613, 522, 1310, 858, 1001, 929, 1610, 1418, 1322, 1499, 1421, 2941, 3300, 3533, 3710, 3957, 2065, 2241, 3685, 4595, 3697, 3930, 5956, 12074, 5509, 5874, 14690, 7968, 6084, 6373, 12413, 12740, 6694, 21878
Offset: 1

Views

Author

Artur Jasinski, Apr 05 2008

Keywords

Examples

			a(3)=13 because 13 is the largest number k such that the equation 5*x_1 + 7*x_2 + 11*x_3 = k has no solution for any nonnegative x_i. (In other words, for every k > 13 there exist one or more solutions.)

Crossrefs

Frobenius numbers for k successive primes: A037165 (k=2), this sequence (k=3), A138990 (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]}], {n, 1, 100}]
FrobeniusNumber/@Partition[Prime[Range[50]],3,1] (* Harvey P. Dale, Dec 01 2015 *)

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 *)

A069755 Frobenius number of the numerical semigroup generated by 3 consecutive triangular numbers.

Original entry on oeis.org

17, 29, 89, 125, 251, 323, 539, 659, 989, 1169, 1637, 1889, 2519, 2855, 3671, 4103, 5129, 5669, 6929, 7589, 9107, 9899, 11699, 12635, 14741, 15833, 18269, 19529, 22319, 23759, 26927, 28559, 32129, 33965, 37961, 40013, 44459, 46739, 51659
Offset: 2

Views

Author

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

Keywords

Comments

The Frobenius number of the 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. Any three successive triangular numbers are relatively prime, so they generate a numerical semigroup with a Frobenius number.

Examples

			a(2)=17 because 17 is not a nonnegative linear combination of 3, 6 and 10 but all numbers greater than 17 are.

Links

Crossrefs

Cf. A000217, A037165, A059769, A069756-A069762.

Programs

  • Mathematica
    tri=Range[40]Range[2,41]/2; Table[t=CoefficientList[Series[1/(1-x^tri[[n]])/(1-x^tri[[n+1]])/(1-x^tri[[n+2]]), {x,0,n(n+1)(n+2)}], x]; Last[Position[t,0]-1][[1]], {n,2,33}] (* T. D. Noe, Nov 27 2006 *)
Rest[FrobeniusNumber/@Partition[Accumulate[Range[50]],3,1]] (* Harvey P. Dale, Oct 04 2011 *)

Formula

Conjectures from Colin Barker, Nov 22 2012: (Start)
a(n) = (-14 + 6*(-1)^n + (3+9*(-1)^n)*n + 3*(5+(-1)^n)*n^2 + 6*n^3)/8.
G.f.: x^2*(17 + 12*x + 9*x^2 - 3*x^4 + x^6) / ((1 - x)^4*(1 + x)^3). (End)
Conjectures from Colin Barker, Mar 21 2017: (Start)
a(n) = (6*n^3 + 18*n^2 + 12*n - 8)/8 for n even.
a(n) = (6*n^3 + 12*n^2 - 6*n - 20)/8 for n odd. (End)

Extensions

Corrected by T. D. Noe, Nov 27 2006

A069764 Frobenius number of the numerical semigroup generated by consecutive octahedral numbers.

Original entry on oeis.org

89, 773, 3611, 12179, 33349, 78889, 167383, 326471, 595409, 1027949, 1695539, 2690843, 4131581, 6164689, 8970799, 12769039, 17822153, 24441941, 32995019, 43908899, 57678389, 74872313, 96140551, 122221399, 153949249, 192262589, 238212323, 292970411, 357838829
Offset: 2

Views

Author

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

Keywords

Comments

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 octahedral 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.

Examples

			a(2)=89 because 89 is not a nonnegative linear combination of 6 and 19 (the second and third octahedral numbers), but all integers greater than 89 are.

Links

Crossrefs

Cf. A005900, A037165, A059769, A069755-A069763.

Programs

  • Mathematica
    FrobeniusNumber/@Partition[Rest[Table[(n(2n^2+1))/3,{n,30}]],2,1] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{89,773,3611,12179,33349,78889,167383},30] (* Harvey P. Dale, Nov 19 2015 *)

Formula

a(n) = ((1/3)*n*(2*n^2+1)-1)*((1/3)*(n+1)*(2*(n+1)^2+1)-1)-1.
G.f.: x^2*(89+150*x+69*x^2+20*x^3-13*x^4+6*x^5-x^6)/(1-x)^7. - Colin Barker, Feb 12 2012
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+ 21*a(n-5)- 7*a(n-6)+a(n-7). - Harvey P. Dale, Nov 19 2015

Extensions

More terms from Carl Najafi, Sep 10 2011

A083553 Product of prime(n+1)-1 and prime(n)-1.

Original entry on oeis.org

2, 8, 24, 60, 120, 192, 288, 396, 616, 840, 1080, 1440, 1680, 1932, 2392, 3016, 3480, 3960, 4620, 5040, 5616, 6396, 7216, 8448, 9600, 10200, 10812, 11448, 12096, 14112, 16380, 17680, 18768, 20424, 22200, 23400, 25272, 26892, 28552, 30616, 32040
Offset: 1

Views

Author

Labos Elemer, May 22 2003

Keywords

Comments

The conductor of x*prime(n) + y*prime(n+1); that is, for all k >= a(n), there exist nonnegative integers x and y such that k = x*prime(n) + y*prime(n+1). - T. D. Noe, Sep 22 2004

Examples

			n=25: a(25) = (97-1)*(101-1) = 9600.

References

  • David Bressoud and Stan Wagon, A Course in Computational Number Theory, Key College Pub., 2000, p. 46.

Links

Crossrefs

Cf. A000040, A006093, A058263, A083538-A083555, A099407 (terms halved), A172042 [= A000010(a(n))], A256617.
One more than A037165.
Column 3 of A379010.

Programs

  • Mathematica
    f[x_] := Prime[x]-1; Table[f[w+1]*f[w], {w, 1, 128}]
  • PARI
    A083553(n) = ((prime(1+n)-1)*(prime(n)-1)); \\ Antti Karttunen, Dec 14 2024

Formula

a(n) = A006093(n+1)*A006093(n) = (prime(n+1)-1)*(prime(n)-1).
a(n) = A037165(n) + 1.
a(n) = 2*A099407(n). - Antti Karttunen, Dec 14 2024
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