A244509 -id:A244509 - cp's OEIS frontend

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

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

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

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