A117762 -id:A117762 - cp's OEIS frontend

A127917 Product of three numbers: n-th prime, previous number, and following number.

6, 24, 120, 336, 1320, 2184, 4896, 6840, 12144, 24360, 29760, 50616, 68880, 79464, 103776, 148824, 205320, 226920, 300696, 357840, 388944, 492960, 571704, 704880, 912576, 1030200, 1092624, 1224936, 1294920, 1442784, 2048256, 2247960, 2571216, 2685480, 3307800

Offset: 1

Artur Jasinski, Feb 06 2007

a(n) is the order of the matrix group SL(2,prime(n)). - Tom Edgar, Sep 28 2015

G. C. Greubel, Table of n, a(n) for n = 1..10000
J. B. Marshall, On the extension of Fermat's theorem to matrices of order n, Proceedings of the Edinburgh Mathematical Society 6 (1939) 85-91. See (11) page 90-91 when p=2.
Index to sequences related to prime powers.

Cf. A036689, A034953, A084920, A117762.

Cf. A065470, A065487.

[6] cat [NthPrime(n)*(NthPrime(n)^2-1): n in [2..40]]; // Vincenzo Librandi, Sep 29 2015

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1), {n, 1, 100}]
Table[p(p^2-1),{p,Prime[Range[40]]}] (* Harvey P. Dale, Apr 26 2025 *)

forprime(p=2,1e3,print1(6*binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = prime(n)*(prime(n)^2-1);
vector(40, n, a(n)) \\ Altug Alkan, Sep 28 2015

a(n) = prime(n)*(prime(n)^2-1). - Tom Edgar, Sep 28 2015
a(n) = 2 * A117762(n), for n > 1. - Altug Alkan, Sep 28 2015
From Amiram Eldar, Nov 22 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065487.
Product_{n>=1} (1 - 1/a(n)) = A065470. (End)

A127918 Half of product of three numbers: n-th prime, previous and following number.

Original entry on oeis.org

3, 12, 60, 168, 660, 1092, 2448, 3420, 6072, 12180, 14880, 25308, 34440, 39732, 51888, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 285852, 352440, 456288, 515100, 546312, 612468, 647460, 721392, 1024128, 1123980, 1285608

Offset: 1

Artur Jasinski, Feb 06 2007

Apart from the first term, the same as A117762. - R. J. Mathar, Jun 14 2008
Except the first term, a(n) is the area of the integer-sided isosceles triangle ABC with AB=AC such that the altitude AH is of prime(n) length.
The couples (a(n), altitude) are (12,3), (60,5), (168,7), (660,11), (1092,13), ... and the sequence of the ratio a(n)/prime(n) is {4, 12, 24, 60, 84, 144, 180, ...} - see A084921. - Michel Lagneau, Oct 23 2013
a(n) is also equal to the number of reducible quadratic polynomials in the field of size prime(n). - James East, Apr 26 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A036689, A034953, A127917, A127920.

[(NthPrime(n)+1)*NthPrime(n)*(NthPrime(n)-1)/2: n in [1..40]]; // Vincenzo Librandi, Apr 09 2017

Table[(Prime[n] + 1) Prime[n](Prime[n] - 1)/2, {n, 1, 100}]

forprime(p=2,1e3,print1(3*binomial(p+1,3)", ")) \\ Charles R Greathouse IV, Jun 16 2011

A335000 Orders of the groups PSL(m,q) in increasing order as q runs through the prime powers (with repetitions).

Original entry on oeis.org

6, 12, 60, 60, 168, 168, 360, 504, 660, 1092, 2448, 3420, 4080, 5616, 6072, 7800, 9828, 12180, 14880, 20160, 20160, 25308, 32736, 34440, 39732, 51888, 58800, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 262080, 265680, 285852, 352440, 372000, 456288, 515100, 546312

Offset: 1

Michel Marcus, May 19 2020

nonn

60 is the order of PSL(2,4) and of PSL(2,5).
168 is the order of PSL(2,7) and of PSL(3,2).
20160 is the order of PSL(4,2) and of PSL(3,4).
Other repetitions > 20160 for PSL(m,q) groups are not known.
See A334884 and A334994 for variations of this sequence.

			a(5) = #PSL(2,7) = (7^2-1)*7/gcd(2,6) = 168, and,
a(6) = #PSL(3,2) = (2^3-1)*(2^3-2)*2^2/gcd(3,1) = 168.

Wikipedia, Projective linear group.
Index entries for sequences related to groups.

Cf. A002884 \ {1} (PSL(n,2)), A117762 (PSL(2, prime(n))).

Cf. A334884 (another case with repetitions), A334994 (without repetitions).

#PSL(m,q) = (Product_{j=0..m-2} (q^m - q^j)) * q^(m-1) / gcd(m,q-1). - Bernard Schott, May 19 2020

A352806 Orders of the finite groups PSL_2(K) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 12, 60, 60, 168, 504, 360, 660, 1092, 4080, 2448, 3420, 6072, 7800, 9828, 12180, 14880, 32736, 25308, 34440, 39732, 51888, 58800, 74412, 102660, 113460, 262080, 150348, 178920, 194472, 246480, 265680, 285852, 352440, 456288, 515100, 546312, 612468, 647460

Offset: 1

Jianing Song, Apr 04 2022

nonn

For a communtative unital ring R, PSL_n(R), the projective special linear group of order n over R, is defined as SL_n(R)/{r*I_n: r^n = 1}. This is related to PGL_n(R), the projective general linear group of order n over R, which is defined as GL_n(R)/{r*I_n: r is a unit of R}.

Note that a(3) = a(4) = 60 refer to the same group (PSL(2,4) = PSL(2,5) = Alt(5)). Also PSL(2,9) = Alt(6).

			a(6) = 504 since A246655(6) = 8, so a(6) = 8*(8^2-1)/gcd(2,8-1) = 504.
a(7) = 360 since A246655(7) = 9, so a(7) = 9*(9^2-1)/gcd(2,9-1) = 360.

Jianing Song, Table of n, a(n) for n = 1..10000
Groupprops, Projective special linear group

Cf. A246655.
Order of GL(2,q): A059238;
SL(2,q): A329119;
PGL(2,q): A329119;
PSL(2,q): this sequence;
Aut(GL(2,q)): A353247;
PGammaL(2,q) = Aut(SL(2,q)) = Aut(PGL(2,q)) = Aut(PSL(2,q)): A352807.
A117762 is a subsequence, A335000 is a supersequence.

[(q+1)*q*(q-1)/gcd(2,q-1) | q <- [1..200], isprimepower(q)]

|PSL(2,q)| = q*(q^2-1)/2 if q is odd, q*(q^2-1) otherwise.
|PSL(2,q)| = |PGL(2,q)|/gcd(2,q-1) = |SL(2,q)|/gcd(2,q-1).
In general, |PSL(n,q)| = |PGL(n,q)|/gcd(n,q-1) = |SL(n,q)|/gcd(n,q-1).

A334884 Order of the non-isomorphic groups PSL(m,q) [or PSL_m(q)] in increasing order as q runs through the prime powers.

Original entry on oeis.org

6, 12, 60, 168, 360, 504, 660, 1092, 2448, 3420, 4080, 5616, 6072, 7800, 9828, 12180, 14880, 20160, 20160, 25308, 32736, 34440, 39732, 51888, 58800, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 262080, 265680, 285852, 352440, 372000, 456288

Offset: 1

Bernard Schott, May 14 2020

nonn

The projective special linear group PSL(m,q) is the quotient group of SL(m,q) with its center.
Theorem: The group PSL(m,q) is simple except for PSL(2,2) and PSL(2,3).
Exceptional isomorphisms (let "==" denote "isomorphic to"):
a(1) = 6 for PSL(2,2) == GL(2,2) == SL(2,2) == S_3 (see example).
a(2) = 12 for PSL(2,3) == A_4.
a(3) = 60 for PSL(2,4) and for PSL(2,5) with PSL(2,4) == PSL(2,5) == A_5 that is the smallest nonabelian simple group.
a(4) = 168 for PSL(2,7) and for PSL(3,2) with PSL(2,7) == PSL(3,2); PSL(2, 7) is the second smallest nonabelian simple group (see example).
a(5) = 360 for PSL(2,9) == A_6.
a(18) = a(19) = 20160 for PSL(4,2) == A_8 and for PSL(3,4) non-isomorphic to A_8 (see comment in A137863).
Array for order of PSL(m,q):
m\q|  2         3            4 =2^2            5               7
----------------------------------------------------------------------
2 |   6         12             60              60             168
3 |  168       5616           20160          372000         1876896
4 | 20160    6065280        987033600      7254000000    2317591180800
5 | 9999360 237783237120  258492255436800 56653740000000000  #PSL(5,7)
              with #PSL(5,7) = 187035198320488089600

			a(1) = #PSL(2,2) = (2^2-1)*2 = 6 and the 6 elements of PSL(2,2) that is isomorphic to S_3 are the 6 following 2 X 2 matrices with entries in F_2:
   (1 0)   (1 1)   (1 0)   (0 1)   (0 1)   (1 1)
   (0 1) , (0 1) , (1 1) , (1 0) , (1 1) , (1 0).
a(4) = #PSL(2,7) = (7^2-1)*7/gcd(2,6) = 168, and also,
a(4) = #PSL(3,2) = (2^3-1)*(2^3-2)*2^2/gcd(3,1) = 168.

Wikipedia, Projective linear group

Subsequence: A117762 (PSL(2,prime(n))).
Cf. A137863.
Cf. A334994 and A335000 for other versions of this sequence.

#PSL(m,q) = (Product_{j=0..m-2} (q^m - q^j)) * q^(m-1) / gcd(m,q-1).

A334994 Orders of the groups PSL(m,q) in increasing order as q runs through the prime powers (without repetitions).

Original entry on oeis.org

6, 12, 60, 168, 360, 504, 660, 1092, 2448, 3420, 4080, 5616, 6072, 7800, 9828, 12180, 14880, 20160, 25308, 32736, 34440, 39732, 51888, 58800, 74412, 102660, 113460, 150348, 178920, 194472, 246480, 262080, 265680, 285852, 352440, 372000, 456288, 515100, 546312

Offset: 1

Michel Marcus, May 19 2020

nonn

is the order of PSL(2,4) or PSL(2,5).
is the order of PSL(2,7) or PSL(3,2).
is the order of PSL(4,2) or PSL(3,4).
See A334884 and A335000 for variations of this sequence.

			#PSL(2,7) = (7^2-1)*7/gcd(2,6) = 168 = a(4), and,
#PSL(3,2) = (2^3-1)*(2^3-2)*2^2/gcd(3,1) = 168 = a(4).

Wikipedia, Projective linear group
Index entries for sequences related to groups.

Cf. A117762 (PSL(2, prime(n))).

Cf. A334884 and A335000 (both with repetitions, but different).

#PSL(m,q) = (Product_{j=0..m-2} (q^m - q^j)) * q^(m-1) / gcd(m,q-1). - Bernard Schott, May 19 2020

A270775 a(n) is the number of invertible 2 X 2 upper triangular matrices over Z_p where p = prime(n).

Original entry on oeis.org

2, 12, 80, 252, 1100, 1872, 4352, 6156, 11132, 22736, 27900, 47952, 65600, 75852, 99452, 143312, 198476, 219600, 291852, 347900, 378432, 480636, 558092, 689216, 893952, 1010000, 1071612, 1202252, 1271376, 1417472, 2016252, 2213900, 2533952, 2647116, 3263696

Offset: 1

Tom Edgar, Mar 22 2016

nonn

a(n) divides A244509(n).

			Over Z_2, there are only two invertible upper triangular 2 X 2 matrices: [[1,0],[0,1]] and [[1,1],[0,1]] so a(1) = 2.

Gregor Olsavsky, Groups formed from 2 X 2 matrices over Z_p, Mathematics Magazine, Vol. 63, No. 4 (Oct., 1990), pp. 269-272.

Cf. A244509, A127917, A117762.

[nth_prime(p)*(nth_prime(p)-1)^2 for p in [1..35]]

a(n) = p*(p-1)^2 where p = prime(n).

Sum 1/a(n) = A382552. - R. J. Mathar, Mar 31 2025

A262354 a(n) is the number of 2 X 2 matrices over Z_p with determinant in {1,-1} where p = prime(n).

Original entry on oeis.org

6, 48, 240, 672, 2640, 4368, 9792, 13680, 24288, 48720, 59520, 101232, 137760, 158928, 207552, 297648, 410640, 453840, 601392, 715680, 777888, 985920, 1143408, 1409760, 1825152, 2060400, 2185248, 2449872, 2589840, 2885568, 4096512, 4495920, 5142432, 5370960

Offset: 1

Tom Edgar, Mar 24 2016

nonn

a(n) divides A244509(n).

For n>2 (i.e. p=prime(n)>=5), a(n) gives the order of the largest proper subgroup of GL(2,Z_p).

Gregor Olsavsky, Groups formed from 2 X 2 matrices over Z_p, Mathematics Magazine, Vol. 63, No. 4 (Oct., 1990), pp. 269-272.

Cf. A244509, A127917, A117762, A270775.

Prepend[2 Table[(Prime@ n + 1) Prime@ n (Prime@ n - 1), {n, 2, 34}], 6] (* Michael De Vlieger, Mar 24 2016, after Artur Jasinski at A127917 *)

lista(nn) = {print1(6, ", "); forprime(p=3, nn, print1(2*p*(p^2-1), ", ")); } \\ Altug Alkan, Mar 24 2016

[6] + [2*p*(p^2-1) for p in prime_range(3,150)]

For n>1, a(n) = 2*p*(p^2-1) where p = prime(n).

For n>1, a(n) = 2*A127917(n).

cp's OEIS Frontend

A127917 Product of three numbers: n-th prime, previous number, and following number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A127918 Half of product of three numbers: n-th prime, previous and following number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A335000 Orders of the groups PSL(m,q) in increasing order as q runs through the prime powers (with repetitions).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A352806 Orders of the finite groups PSL_2(K) when K is a finite field with q = A246655(n) elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A334884 Order of the non-isomorphic groups PSL(m,q) [or PSL_m(q)] in increasing order as q runs through the prime powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A334994 Orders of the groups PSL(m,q) in increasing order as q runs through the prime powers (without repetitions).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A270775 a(n) is the number of invertible 2 X 2 upper triangular matrices over Z_p where p = prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Formula

A262354 a(n) is the number of 2 X 2 matrices over Z_p with determinant in {1,-1} where p = prime(n).

Views

Author

Keywords

Comments

Links