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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A100422 Numbers n such that 30*n+{1,7,11,13,17,23,29} are all prime.

Original entry on oeis.org

1, 53887, 114731, 123306, 139742, 210554, 471745, 480859, 619039, 630862, 858929, 1075873, 1306614, 1714945, 1913514, 2767458, 3014285, 3454137, 3518243, 3699151, 3864512, 3874291, 4274376, 4862362, 4878329, 4937822
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004

Keywords

Comments

Values are 1 mod 7.
In each case, the 7 primes are necessarily consecutive. See the comment in A100418. - Peter Munn, Sep 06 2023

Crossrefs

Cf. A005776, A007775, A076205, A100418-A100423.

Programs

  • Magma
    [ n: n in [0..5000000] | forall{ q: q in [1, 7, 11, 13, 17, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 23 2011
  • Maple
    a:= proc(n) option remember;
      local m;
      if n=1 then 1
      else for m from 30*(a(n-1)+7) by 210
           while not (isprime (m+1) and isprime (m+7) and
                 isprime (m+11) and isprime (m+13) and
                 isprime (m+17) and isprime (m+23) and
                 isprime (m+29))
           do od; m/30
        fi
    end:
seq (a(n), n=1..10);
  • Mathematica
    Select[Range[5000000],And@@PrimeQ/@(30(#)+{1,7,11,13,17,23,29})&]  (* Harvey P. Dale, Feb 23 2011 *)

Extensions

Edited by Don Reble, Nov 17 2005

A100421 Numbers n such that 30*n+{1,7,11,13,19,23,29} are all prime.

Original entry on oeis.org

2, 79, 391701, 505017, 740413, 787187, 933025, 1169863, 1333719, 1406792, 2212261, 2719950, 2962738, 3125992, 3284955, 3384586, 3727271, 3821295, 3861881, 4320864, 4439878, 4764356, 5014865, 5480190, 5879274, 6124442
Offset: 1

Views

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004

Keywords

Comments

Values are 2 mod 7.
In each case, the 7 primes are necessarily consecutive. See the comment in A100418. - Peter Munn, Sep 06 2023

Crossrefs

Cf. A005776, A007775, A076205, A100418-A100420, A100422, A100423.

Programs

  • Magma
    [ n: n in [2..70000000 by 7] | forall{ q: q in [1, 7, 11, 13, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011
  • Mathematica
    Select[Range[7*10^6],AllTrue[30#+{1,7,11,13,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 16 2016 *)

Extensions

Edited by Don Reble, Nov 17 2005

A118889 Ratio of Dimensions of the traditional Cartan exceptional group sequence A1,G2,F4,E6,E7,E8 to the Cartan matrix Dimension: Dimc={1, 2, 4, 6, 7, 8} DimG={3, 14, 52, 78, 133, 248} DimG/DimC={3, 7, 13, 13, 19, 31}.

Original entry on oeis.org

3, 7, 13, 13, 19, 31
Offset: 1

Views

Author

Roger L. Bagula, May 17 2007

Keywords

Comments

The sequence is inherently unordered, because there is no standard ordering of these groups. - R. J. Mathar, Dec 04 2011

Crossrefs

Cf. A117133, A129766, A005556, A005763, A005776.

Programs

  • Mathematica
    (* Cartan Matrices: *)
e[3] = {{2}};
e[4] = {{2, -3}, {-1, 2}};
e[5] = {{2, -1, 0, 0}, {-1, 2, -2, 0}, {0, -1, 2, -1}, {0, 0, -1, 2}};
e[6] = {{2, 0, -1, 0, 0, 0}, {0, 2, 0, -1, 0, 0}, {-1, 0, 2, -1, 0, 0}, { 0, -1, -1, 2, -1, 0}, { 0, 0, 0, -1, 2, -1}, { 0, 0, 0, 0, -1, 2}};
e[7] = {{2, 0, -1, 0, 0, 0, 0}, {0, 2, 0, -1, 0, 0, 0}, {-1, 0, 2, -1, 0, 0, 0}, {0, -1, -1, 2, -1, 0, 0}, {0, 0, 0, -1, 2, -1, 0}, { 0, 0, 0, 0, -1, 2, -1 }, { 0, 0, 0, 0, 0, -1, 2 }};
e[8] = { {2, 0, -1, 0, 0, 0, 0, 0}, { 0, 2, 0, -1, 0, 0, 0, 0}, {-1, 0, 2, -1, 0, 0, 0, 0}, {0, -1, -1, 2, -1, 0, 0, 0}, {0, 0, 0, -1, 2, -1, 0, 0}, { 0, 0, 0, 0, -1, 2, -1, 0}, { 0, 0, 0, 0, 0, -1, 2, -1}, {0, 0, 0, 0, 0, 0, -1, 2}} ;
a0 = Table[Length[CoefficientList[CharacteristicPolynomial[e[n], x], x]] - 1, {n, 3, 8}]; (* Poincaré Polynomials*)
(*Poincaré polynomial exponents for G2, E6, E7, E8 from A005556, A005763, A005776 and Armand Borel's Essays in History of Lie Groups and Algebraic Groups*) (* b[n] = a[n] + 1 : DimGroup = Apply[Plus, b[n]]*)
a[0] = {1};
a[1] = {1, 5};
a[2] = {1, 5, 7, 11};
a[3] = {1, 4, 5, 7, 8, 11};
a[4] = {1, 5, 7, 9, 11, 13, 17};
a[5] = {1, 7, 11, 13, 17, 19, 23, 29};
b0 = Table[Length[CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1, Length[a[i]]}]], t]] - 1, {i, 0, 5}];
Table[b0[[n]]/a0[[n]], {n, 1, Length[a0]}]

Formula

P[n]=Poincare-Polynomial[n]=Product[1+t^A129766[m],{m,1,n}]
DimG[n]=Length[CoefficientList[P[n],t]]-1
Pc[n]=CharacteristicPolynomial[M[n],x]
DimC[n]=Length[CoefficientList[Pc[n],x]]-1
a[n]=DimG[n]/DimC[n]

A124680 Heights of irreducible representations of E_8.

Original entry on oeis.org

58, 92, 114, 136, 168, 182, 220, 270
Offset: 1

Views

Author

Roger L. Bagula, Dec 25 2005

Keywords

Examples

			The factored sequence, [2*29, 2^2*23, 2*3*19, 2^3*17, 2^3*3*7, 2*7*13, 2^2*5*11, 2*3^3*5], shows a close relationship to A005776. - _N. J. A. Sloane_, Dec 25 2006

References

  • R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

Crossrefs

Cf. A005776, A030650, A121732.

A129769 Exponents m(i) for exceptional groups with best guesses for E7 1/2 and E9 added (there is a problem with the dimension of E9 as no sum of odd numbers will equal the 484, I get 483): triangular sequence is: A1,G2,F4,E6,E7 E7 1/2,E8,E9.

Original entry on oeis.org

1, 1, 5, 1, 5, 7, 11, 1, 4, 5, 7, 8, 11, 1, 5, 7, 9, 11, 13, 17, 1, 6, 9, 11, 13, 15, 17, 19, 1, 7, 11, 13, 17, 19, 23, 29, 1, 11, 17, 19, 23, 29, 31, 51, 55
Offset: 1

Views

Author

Roger L. Bagula, May 16 2007

Keywords

Comments

Betti number row sums: Table[Apply[Plus, CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1, Length[a[i]]}]], t]], {i, 0, 7}] {2, 4, 16, 64, 128, 256, 256, 512} Group dimensions sums: b[n_] = 2*a[n] + 1 Table[Apply[Plus, b[n]], {n, 0, 7}] {3, 14, 52, 78, 133, 190, 248, 483}.
From these exponents it is possible to get Poincaré polynomial estimates for the new E7 1/2 and E8 that best fit the pattern of the known exponents.

References

  • J. M. Landsberg, The sextonions and E_{7 1/2} (with L.Manivel) (Advances in Math 201(2006) p143 - 179) page 22; J. M. Landsberg, http://www.math.tamu.edu/~jml/LMsexpub.pdf: The sextonions and E_{7 1/2}
  • Armand Borel's Essays in History of Lie Groups and Algebraic Groups: gives G2 PoincarĂ© polynomial, History of Mathematics, V. 21; http://www.amazon.com/Essays-History-Groups-Algebraic-Mathematics/dp/0821802887/ref=pd_rhf_p_3/104-0029617-0633535

Crossrefs

Cf. A005556, A005763, A005776.

Programs

  • Mathematica
    a[0] = {1}; a[1] = {1, 5}; a[2] = {1, 5, 7, 11}; a[3] = {1, 4, 5, 7, 8, 11}; a[4] = {1, 5, 7, 9, 11, 13, 17}; a[5] = {1, 6, 9, 11, 13, 15, 17, 19}; a[6] = {1, 7, 11, 13, 17, 19, 23, 29}; a[7] = {1, 11, 17, 19, 23, 29, 31, 51, 55};

Formula

a(0) = {1}; a(1) = {1, 5}; a(2) = {1, 5, 7, 11}; a(3) = {1, 4, 5, 7, 8, 11}; a(4) = {1, 5, 7, 9, 11, 13, 17}; a(5) = {1, 6, 9, 11, 13, 15, 17, 19}; a(6) = {1, 7, 11, 13, 17, 19, 23, 29}; a(7) = {1, 11, 17, 19, 23, 29, 31, 51, 55};
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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