A005776 -id:A005776 - cp's OEIS frontend

A129766 Triangular array read by rows, made up of traditional exceptional groups plus A1: as A1,G2,F4,E6,E7,E8 as m(i) exponents as in A005556, A005763, A005776.

1, 1, 5, 1, 5, 7, 11, 1, 4, 5, 7, 8, 11, 1, 5, 7, 9, 11, 13, 17, 1, 7, 11, 13, 17, 19, 23, 29

Offset: 1

Roger L. Bagula, May 16 2007

Extra condition of group dimension: b[n] = a[n] + 1 ; DimGroup = Apply[Plus, b[n]]; Table[Apply[Plus, b[n]], {n, 0, 5}] {3, 14, 52, 78, 133, 248} Extra condition of Betti sum: Table[Apply[Plus, CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1,Length[a[i]]}]], t]], {i, 0, 5}] {2, 4, 16, 64, 128, 256} These exponents are necessary to the Poincaré polynomials for these exceptional groups.

			1;
1,5;
1,5,7,11;
1,4,5,7,8,11;
1,5,7,9,11,13,17;
1,7,11,13,17,19,23,29;

Armand Borel, Essays in History of Lie Groups and Algebraic Groups gives G2 Poincaré polynomial, History of Mathematics, V. 21.

Cf. A005556, A005763, A005776.

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}; Flatten[Table[a[n], {n, 0, 5}]]

A100418 Numbers k such that 30*k + {1,11,13,17,19,23,29} are all prime.

Original entry on oeis.org

49, 34083, 41545, 48713, 140609, 524027, 616812, 855281, 1314397, 1324750, 1636152, 2281293, 2927134, 3401412, 3605413, 4989341, 5212221, 5284979, 5406303, 5645269, 6141254, 6342728, 7231434, 7347697, 7637329, 8027068, 8161657, 8372756, 8392776, 8567216, 8986096, 9145563

Offset: 1

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

Values are 0 mod 7.
From Peter Munn, Sep 06 2023: (Start)
In each case, the 7 primes are necessarily consecutive.
As A065706 demonstrates, many intervals of 27 integers contain 8 primes, but only A364678(30) = 7 primes can occur between adjacent positive multiples of 30. This is because there are 8 values {1,7,11,13,17,19,23,29} coprime to 30, but they cover every residue class modulo 7, which means at least one of 30*k + {1,7,11,13,17,19,23,29} is divisible by 7.
1 and 29 are in the same residue class, but if we remove any of the other coprime integers there is a class that is not represented in the set. For this sequence, we remove 7, so when k is a multiple of 7, none of 30*k + {1,11,13,17,19,23,29} is a multiple of 2, 3, 5 or 7 and the set can potentially be 7 consecutive primes.
The sequences for the other appropriate subsets of 7 coprime values are A100419-A100423.
(End)

David A. Corneth, Table of n, a(n) for n = 1..10309

Cf. A005776, A007775, A056956, A065706, A076205, A100419-A100423, A364678.

[ n: n in [0..70000000 by 7] | forall{ q: q in [1, 11, 13, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[803*10^4],AllTrue[30#+{1,11,13,17,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 11 2019 *)

{pav7(mx)= local(wp=[1,11,13,17,19,23,29],v=[],i,j,m); for(k=1,mx, i=k*30;j=1;m=1;while(m&&(j<8),m=isprime(i+wp[j]);j+=1);if(m,v=concat(v,k))); return(v)}

Edited by Don Reble, Nov 17 2005

A100423 Numbers n such that 30*n+{1,7,11,13,17,19,29} are all prime.

Original entry on oeis.org

62, 188, 9491, 31982, 38226, 38520, 89459, 168237, 175125, 368248, 471078, 634892, 704416, 803102, 994748, 1436315, 1488857, 1605484, 1842553, 1945824, 2282958, 2465266, 2620715, 2627029, 2705037, 4282305, 5569899, 5914824

Offset: 1

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

Values are 6 mod 7.

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

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

[ n: n in [0..6000000] | forall{ q: q in [1, 7, 11, 13, 17, 19, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[6*10^6],AllTrue[30#+{1,7,11,13,17,19,29},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 21 2021 *)

Edited by Don Reble, Nov 17 2005

A076205 Numbers n such that 30*n+{1,7,11,13,17,19,23,29} are all composite.

Original entry on oeis.org

360, 523, 654, 941, 1020, 1047, 1064, 1136, 1188, 1213, 1264, 1280, 1343, 1355, 1445, 1477, 1515, 1526, 1530, 1533, 1582, 1623, 1652, 1693, 1842, 1900, 1960, 2018, 2039, 2129, 2208, 2280, 2309, 2332, 2406, 2413, 2440, 2499, 2539, 2622, 2633, 2650, 2657

Offset: 1

Donald S. McDonald, Nov 02 2002

H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 141.

Klaus Brockhaus, Table of n, a(n) for n = 1..6829 (terms < 100000)

Cf. A005776, A007775, A100418-A100423.

[ n: n in [0..3000] | forall{ q: q in [1, 7, 11, 13, 17, 19, 23, 29] | not IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[3000],AllTrue[30#+{1,7,11,13,17,19,23,29},CompositeQ]&] (* Harvey P. Dale, Jan 06 2022 *)

{cav(mx)= local(wp=[1,7,11,13,17,19,23,29],v=[],i,j,m); for(k=1,mx, i=k*30;j=1;m=1;while(m&&(j<9),m=!isprime(i+wp[j]);j+=1);if(m,v=concat(v,k))); return(v)}

More terms from Ferenc Adorjan (fadorjan(AT)freemail.hu), Nov 19 2004
Edited by Don Reble, Nov 17 2005
Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar

A106403 Primitive exponents of the Weyl group W(E_8).

Original entry on oeis.org

3, 15, 23, 27, 35, 39, 47, 59

Offset: 1

N. J. A. Sloane, May 30 2005

C. Chevalley, The Betti numbers of the exceptional Lie groups, pp. 21-24 of Proc. Internat Congress Math., Cambridge 1950, Amer. Math. Soc., 1952.

Cf. A106373, A106374, A106403, A005556, A005763.

Equals 2*A005776(n) + 1.

A100419 Numbers k such that 30*k+{1,7,13,17,19,23,29} are all prime.

Original entry on oeis.org

89, 6627, 18674, 223949, 229269, 240007, 267356, 606681, 638454, 771496, 951060, 1068030, 1150693, 1254839, 1688923, 1920084, 2413577, 2433289, 2649414, 3053398, 3080572, 3337444, 3586658, 3604256, 3830335, 4137166

Offset: 1

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

nonn

Values are 5 mod 7.

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

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..2020 from Robert Israel)

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

[ n: n in [5..70000000 by 7] | forall{ q: q in [1, 7, 13, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

filter:= proc(n) local j; andmap(isprime, [seq(30*n+j,j=[1,7,13,17,19,23,29])]) end proc:
select(filter, [seq(i,i=5..5*10^6,7)]); # Robert Israel, Nov 04 2024

Select[Range[42*10^5],AllTrue[30#+{1,7,13,17,19,23,29},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 10 2018 *)

Edited by Don Reble, Nov 17 2005

A106373 Primitive exponents of the Weyl group W(E_6).

Original entry on oeis.org

3, 9, 11, 15, 17, 23

Offset: 1

N. J. A. Sloane, May 30 2005

C. Chevalley, The Betti numbers of the exceptional Lie groups, pp. 21-24 of Proc. Internat Congress Math., Cambridge 1950, Amer. Math. Soc., 1952.

Cf. A106374, A106403, A005556, A005763, A005776.

Equals 2*A005556(n) + 1.

A106374 Primitive exponents of the Weyl group W(E_7).

Original entry on oeis.org

3, 11, 15, 19, 23, 27, 35

Offset: 1

N. J. A. Sloane, May 30 2005

C. Chevalley, The Betti numbers of the exceptional Lie groups, pp. 21-24 of Proc. Internat Congress Math., Cambridge 1950, Amer. Math. Soc., 1952.

Cf. A106373, A106403, A005556, A005763, A005776.

Equals 2*A005763(n) + 1.

A089010 a(n) = 1 if n is an exponent of the Weyl group W(E_8), 0 otherwise.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Paul Boddington, Nov 03 2003

The exponents are 1, 7, 11, 13, 17, 19, 23, 29. The point of this sequence is that a similar generating function gives the exponents for any finite Coxeter group.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions

Cf. A005776, A089011.

PadRight[CoefficientList[Series[x(1-x^20)(1-x^24)/((1-x^6)(1-x^10)),{x,0,120}],x],120,0] (* Harvey P. Dale, May 15 2018 *)

Vec(x*(1-x^20)*(1-x^24)/((1-x^6)*(1-x^10)) + O(x^90)) \\ Michel Marcus, Aug 19 2015

G.f.: x*(1-x^20)*(1-x^24)/((1-x^6)*(1-x^10)).

A100420 Numbers n such that 30*n+{1,7,11,17,19,23,29} are all prime.

Original entry on oeis.org

22621, 103205, 149125, 237794, 288467, 321451, 364921, 373370, 404002, 851099, 985933, 1106235, 1594044, 1696874, 1780265, 1824421, 1851756, 2249881, 3112939, 3257538, 3397608, 3601651, 3747356, 4347340, 4710990, 4886284

Offset: 1

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

Values are 4 mod 7.

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

Cf. A005776, A007775, A076205, A100418, A100419, A100421-A100423.

[ n: n in [4..70000000 by 7] | forall{ q: q in [1, 7, 11, 17, 19, 23, 29] | IsPrime(30*n+q) } ]; // Klaus Brockhaus, Feb 24 2011

Select[Range[5000000],And@@PrimeQ[30 #+{1,7,11,17,19,23,29}]&]  (* Harvey P. Dale, Mar 06 2011 *)

Edited by Don Reble, Nov 17 2005

cp's OEIS Frontend

A129766 Triangular array read by rows, made up of traditional exceptional groups plus A1: as A1,G2,F4,E6,E7,E8 as m(i) exponents as in A005556, A005763, A005776.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A100418 Numbers k such that 30*k + {1,11,13,17,19,23,29} are all prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A100423 Numbers n such that 30*n+{1,7,11,13,17,19,29} are all prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Extensions

A076205 Numbers n such that 30*n+{1,7,11,13,17,19,23,29} are all composite.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A106403 Primitive exponents of the Weyl group W(E_8).

Views

Author

Keywords

References

Crossrefs

Formula

A100419 Numbers k such that 30*k+{1,7,13,17,19,23,29} are all prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Extensions

A106373 Primitive exponents of the Weyl group W(E_6).

Views

Author

Keywords

References

Crossrefs

Formula

A106374 Primitive exponents of the Weyl group W(E_7).

Views

Author

Keywords

References

Crossrefs

Formula

A089010 a(n) = 1 if n is an exponent of the Weyl group W(E_8), 0 otherwise.

Views