A049315 -id:A049315 - cp's OEIS frontend

A182609 Number of conjugacy classes in GL(n,19).

1, 18, 360, 6840, 130302, 2475720, 47045520, 893864520, 16983555840, 322687560618, 6131066120640, 116490256285320, 2213314916460120, 42052983412605480, 799006685733239040, 15181127028931412160, 288441413566677788022, 5480386857766875373560

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182604, A182605, A182606, A182607, A182608, A182610, A182611, A182612.

/* The program does not work for n>4: */ [1] cat [NumberOfClasses(GL(n, 19)) : n in [1..4]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*19^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2012

b[n_] := Sum[EulerPhi[d]*19^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-19*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-19*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

MAGMA code edited by Vincenzo Librandi, Jan 24 2013

A182612 Number of conjugacy classes in GL(n,27).

Original entry on oeis.org

1, 26, 728, 19656, 531414, 14348152, 387419760, 10460332792, 282429516096, 7625596933890, 205891131543552, 5559060551656248, 150094635282119528, 4052555152616676888, 109418989131110078784, 2954312706539971597184, 79766443076861647780830

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182604, A182605, A182606, A182607, A182608, A182609, A182610, A182611.

/* The program does not work for n>4: */ [1] cat [ NumberOfClasses(GL(n, 27)) : n in [1..4] ];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*27^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 03 2012

b[n_] := Sum[EulerPhi[d]*27^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-27*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-27*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

MAGMA code edited by Vincenzo Librandi, Jan 24 2013

A182610 Number of conjugacy classes in GL(n,23).

Original entry on oeis.org

1, 22, 528, 12144, 279818, 6435792, 148035360, 3404812752, 78310972608, 1801152369478, 41426510921664, 952809751186128, 21914624425304688, 504036361781716368, 11592836324384010432, 266635235460831961152, 6132610415677439376122, 141050039560581098947824

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182604, A182605, A182606, A182607, A182608, A182609, A182611, A182612.

/* The program does not work for n>4: */ [1] cat [NumberOfClasses(GL(n, 23)) : n in [1..4]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*23^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 03 2012

b[n_] := Sum[EulerPhi[d]*23^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-23*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-23*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

MAGMA code edited by Vincenzo Librandi, Jan 24 2013

A182611 Number of conjugacy classes in GL(n,25).

Original entry on oeis.org

1, 24, 624, 15600, 390600, 9764976, 244140000, 6103499376, 152587874400, 3814696859400, 95367431234400, 2384185780844400, 59604644765235024, 1490116119130470000, 37252902984364860000, 931322574609121110624, 23283064365380605500600, 582076609134515127375600

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182604, A182605, A182606, A182607, A182608, A182609, A182610, A182612.

/* The program does not work for n>4: */ [1] cat [NumberOfClasses(GL(n, 25)) : n in [1..4]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*25^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2012

b[n_] := Sum[EulerPhi[d]*25^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-25*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-25*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

MAGMA code edited by Vincenzo Librandi, Jan 23 2013

A221578 A sum over partitions (q=6), see first comment.

Original entry on oeis.org

1, 5, 35, 210, 1290, 7735, 46620, 279685, 1679370, 10076190, 60464670, 362787810, 2176773305, 13060638360, 78364108620, 470184650495, 2821109573550, 16926657432510, 101559954663930, 609359727929610, 3656158427989830, 21936950567886270, 131621703769781995

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=6 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P = [p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..400

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*6^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 24 2013

b[n_] := Sum[EulerPhi[d]*6^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-6*x^n)  );
v=Vec(gf)

A221579 A sum over partitions (q=10), see first comment.

Original entry on oeis.org

1, 9, 99, 990, 9990, 99891, 999900, 9998901, 99998910, 999989010, 9999989010, 99999889110, 999999890109, 9999998890200, 99999998891100, 999999988901199, 9999999988902090, 99999999888912090, 999999999889011990, 9999999998889021990

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=10 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P = [p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..500

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*10^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 25 2013

b[n_] := Sum[EulerPhi[d]*10^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-10*x^n)  );
v=Vec(gf)

A221580 A sum over partitions (q=12), see first comment.

Original entry on oeis.org

1, 11, 143, 1716, 20724, 248677, 2985840, 35829937, 429979836, 5159757900, 61917341772, 743008099548, 8916100178843, 106993202123808, 1283918461295184, 15407021535521759, 184884258855973380, 2218611106271412996, 26623333280416468596, 319479999364994391924

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=12 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P = [p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..300

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*12^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*12^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-12*x^n)  );
v=Vec(gf)

A221581 A sum over partitions (q=14), see first comment.

Original entry on oeis.org

1, 13, 195, 2730, 38402, 537615, 7529340, 105410565, 1475786130, 20661005638, 289254613830, 4049564590890, 56693911799265, 793714765148760, 11112006817455180, 155568095444334495, 2177953337695895942, 30491346727741970070, 426878854209048054450

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=14 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P = [p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..300

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*14^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*14^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-14*x^n)  );
v=Vec(gf)

A221582 A sum over partitions (q=15), see first comment.

Original entry on oeis.org

1, 14, 224, 3360, 50610, 759136, 11390400, 170855776, 2562887040, 38443305390, 576650336640, 8649755046240, 129746337080864, 1946195056159200, 29192926013193600, 437893890197853824, 6568408355529888210, 98526125332947516960, 1477891880032655307360

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=15 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P = [p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..300

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*15^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*15^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-15*x^n)  );
v=Vec(gf)

A221583 A sum over partitions (q=18), see first comment.

Original entry on oeis.org

1, 17, 323, 5814, 104958, 1889227, 34011900, 612213877, 11019954438, 198359179578, 3570467115834, 64268408079198, 1156831379431973, 20822964829665048, 374813367546080412, 6746640615829343087, 121439531095946141922, 2185911559727028566514

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=18 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products Product_{k=1..L} f(m_k) where L is the number of different parts in the partition P=[p_1^m_1, p_2^m_2, ..., p_L^m_L].
Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":
  q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,
  q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,
  q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.
Sequences where q is not a prime power:
  q=6: A221578, q=10: A221579, q=12: A221580,
  q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..300

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*18^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*18^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-18*x^n)  );
v=Vec(gf)

cp's OEIS Frontend

A182609 Number of conjugacy classes in GL(n,19).

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A182612 Number of conjugacy classes in GL(n,27).

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A182610 Number of conjugacy classes in GL(n,23).

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A182611 Number of conjugacy classes in GL(n,25).

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A221578 A sum over partitions (q=6), see first comment.

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A221579 A sum over partitions (q=10), see first comment.

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A221580 A sum over partitions (q=12), see first comment.

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