A127917 -id:A127917 - cp's OEIS frontend

A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

1, 10, 6, 33, 24, 24, 88, 48, 72, 48, 145, 120, 120, 120, 120, 330, 144, 240, 198, 240, 144, 385, 336, 336, 336, 336, 336, 336, 736, 384, 576, 384, 672, 384, 576, 384, 945, 648, 648, 864, 648, 648, 864, 648, 648, 1450, 720, 1200, 720, 1200, 870, 1200, 720, 1200, 720

Offset: 1

Erdos Pal, Oct 03 2010

The n-th row is {T(n,0),T(n,1),...,T(n,n-1)}.
Let m denote the prime power p^e.
T(m,0) = A020478(m) = (p^(e+1) + p^e-1)*p^(2*e-1).
T(m,1) = A000056(m) = (p^2-1)*p^(3*e-2).
T(prime(n),1) = A127917(n).
Sum_{k=1..n-1} T(n,k) = A005353(n).
T(n,1) = n*A007434(n) for n>=1 because A000056(n) = n*Jordan_Function_J_2(n).
T(2^n,1) = A083233(n) = A164640(2n) for n>=1. Proof: a(n):=T(2^n,1); a(1)=6, a(n)=8*a(n-1); A083233(1)=6 and A083233(n) is a geometric series with ratio 8 (because of its g.f.), too; A164640 = {b(1)=1, b(2)=6, b(n)=8*b(n-2)}.
T(2^n,0) = A165148(n) for n>=0, because  2*T(2^n,0) = (3*2^n-1)*4^n.
T(2^e,2) = A003951(e) for 2 <= e. Proof: T(2^e,2) = 9*8^(e-1) is a series with ratio 8 and initial term 72, as A003951(2...inf) is.
Working with consecutive powers of a prime p, we need a definition (0 <= i < e):
N(p^e,i):=#{k: 0 < k < p^e, gcd(k,p^e) = p^i} = (p-1)*p^(e-1-i). We say that these k's belong to i (respect to p^e). Note that N(p^e,0) = EulerPhi(p^e), and if 0 < k < p^e then gcd(k,p^e) = gcd(k,p^(e+1)). Let T(p^e,[i]) denote the common value of T(p^e,k)'s, where k's belong to i (q.v.PROGRAM); for example, T(p^e,[0]) = T(p^e,1). The number of the 2 X 2 matrices over Z(p^e), T(p^e,0) + Sum_{i=0..e-1} T(p^e,[i])*N(p^e,i) = p^(4e) will be useful.
On the hexagon property: Let prime p be given and let T(p^e,[0]), T(p^e,[1]), T(p^e,[2]), ..., T(p^e,[e-2]), T(p^e,[e-1]) form the e-th row of a Pascal-like triangle, e>=1. Let denote X(r,s) an element of the triangle and its value T(p^r,[s]). Let positive integers a and b given, so that the entries A(m-a,n-b), B(m-a,n), C(m,n+a), D(m+b,n+a), E(m+b,n), F(m,n-b) of the triangle form a hexagon spaced around T(p^m,[n]); if a=b=1 then they surround it. If A*C*E = B*D*F, then we say that the triangle T(.,.) has the "hexagon property". (In the case of binomial coefficients X(r,s) = COMB(r,s), the "hexagon property" holds (see [Gupta]) and moreover gcd(A,C,E) = gcd(B,D,F) (see [Hitotumatu & Sato]).)
Corollary 2.2 in [Brent & McKay] says that, for the d X d matrices over Z(p^e), (mutatis mutandis) T_d(p^e,0) = K*(1-P(d+e-1)/P(e-1)) and T_d(p^e,[i]) = K*(q^e)*((1-q^d)/(1-q))*P(d+i-1)/P(i), where q=1/p, K=(p^e)^(d^2), P(t) = Product_{j=1..t} (1-q^j), P(0):=1. (For the case d=2, we have T(p^e,[i]) = (p+1)*(p^(i+1)-1)*p^(3*e-i-2).) Due to [Brent & McKay], it can be simply proved that for d X d matrices the "hexagon property" is true. The formulation implies an obvious generalization: For the entries A(r,u), B(r,v), C(s,w), D(t,w), E(t,v), F(s,u) of the T_d(.,.)-triangle, a hexagon-like property A*C*E = B*D*F holds. This is false in general for the COMB(.,.)-triangle.
Another (rotated-hexagon-like) property: for the entries A(m-b1,n), B(m-a1,n+c2), C(m+a2,n+c2), D(m+b2,n), E(m+a2,n-c1), F(m-a1,n-c1) of the T_d(.,.)-triangle, the property A*C*E = B*D*F holds, if and only if 2*(a1 + a2) = b1 + b2. This is also in general false for COMB(.,.)-triangle.

			From _Andrew Howroyd_, Jul 16 2018: (Start)
Triangle begins:
    1;
   10,   6;
   33,  24,  24;
   88,  48,  72,  48;
  145, 120, 120, 120, 120;
  330, 144, 240, 198, 240, 144;
  385, 336, 336, 336, 336, 336, 336;
  736, 384, 576, 384, 672, 384, 576, 384;
  945, 648, 648, 864, 648, 648, 864, 648, 648;
  ... (End)

Erdos Pal, Rows n=1..100 of triangle, flattened
Richard P. Brent and Brendan D. McKay, Determinants and ranks of random matrices over Z_m, Discrete Mathematics 66 (1987) pp. 35-49.
A. K. Gupta, Generalized hidden hexagon squares, The Fibonacci Quarterly, Vol 12, Number 1, Feb.1974, pp. 45-46.
S. Hitotumatu, D. Sato, Star of David theorem (I), The Fibonacci Quarterly, Vol 13, Number 1, Feb.1975, p. 70.

Column k=0 is A020478.
Column k=1 is A000056.
Row sums are A005353.
Cf. A000252, A127917.

  (* computing T(p^e,k) ; p=prime, 1<=e, 0<=k

S(p,e)={my(u=vector(p^e)); my(t=(p-1)*p^(e-1)); u[1] = p^e + e*t; for(j=1, p^e-1, u[j+1] = t*(1+valuation(j, p))); vector(#u, k, sum(j=0, #u-1, u[j + 1]*u[(j+k-1) % #u + 1]))}
T(n)={my(f=factor(n), v=vector(n,i,1)); for(i=1, #f~, my(r=S(f[i,1], f[i,2])); for(j=0, #v-1, v[j + 1] *= r[j % #r + 1])); v}
for(n=1, 10, print(T(n))); \\ Andrew Howroyd, Jul 16 2018

T(a*b,k) = T(a,(k mod a))*T(b,(k mod b)) if gcd(a,b) = 1.

Sum_{k=1..n-1, gcd(k,n)=1} T(n,k) = A000252(n). - Andrew Howroyd, Jul 16 2018

Terms a(24)-a(55) from b-file by Andrew Howroyd, Jul 16 2018

A141535 An eighth of the product of three integers surrounding the (n+1)-st prime, minus half of the product of the 3 numbers surrounding n+1.

Original entry on oeis.org

0, 3, 12, 105, 168, 444, 603, 1158, 2550, 3060, 5469, 7518, 8568, 11292, 16563, 23217, 25458, 34167, 40740, 43998, 56307, 65391, 81210, 106272, 120000, 126750, 142155, 149685, 166863, 241152

Offset: 1

Roger L. Bagula, Aug 12 2008

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A127917 := proc(n) p := ithprime(n) ; (p-1)*p*(p+1) ; end: A027480 := proc(n) n*(n+1)*(n+2)/2 ; end: A := proc(n) A127917(n+1)/8-A027480(n) ; end: for n from 1 to 40 do printf("%d,",A(n)) ; od: # R. J. Mathar, Aug 20 2008

a[n_] = (Prime[n + 1] - 1)*Prime[n + 1]*(Prime[n + 1] + 1)/8 - n*(n + 1)*(n + 2)/2; Table[a[n], {n, 1, 30}]
f[n_]:=Module[{pr=Prime[n+1],n1=n+1},(pr(pr^2-1))/8-(n1(n1^2-1))/2]; Array[f,30] (* Harvey P. Dale, May 24 2012 *)

a(n) = A127917(n+1)/8-A027480(n) = {p(n+1)-1}*p(n+1)*{p(n+1)+1}/8-n(n+1)(n+2)/2.

Edited by N. J. A. Sloane, Aug 23 2008

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

A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

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A141535 An eighth of the product of three integers surrounding the (n+1)-st prime, minus half of the product of the 3 numbers surrounding n+1.

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

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Mathematica

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