A306192 -id:A306192 - cp's OEIS frontend

A318173 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n) and whose first column consists of prime(1), prime(n + 1), ..., prime(2*n - 1).

2, -11, 158, -6513, 202790, -12710761, 578257422, -45608219247, 8774909485920, -579515898830751, 115918088707226940, -16737522590543449641, 1282860173728469083872, -189053227741259934603831, 55171097827950314187327460, -16235234399834578732807710581

Offset: 1

Stefano Spezia, Aug 20 2018

sign

The trace of the matrix M(n) is A005843(n).
The sum of the first row of the matrix M(n) is A007504(n).
The permanent of the matrix M(n) is A306457(n).
For n > 1, the subdiagonal sum of the matrix M(n) is A306192(n).

			For n = 1 the matrix M(1) is
   2
with determinant Det(M(1)) = 2.
For n = 2 the matrix M(2) is
   2, 3
   5, 2
with Det(M(2)) = -11.
For n = 3 the matrix M(3) is
   2, 3, 5
   7, 2, 3
  11, 7, 2
with Det(M(3)) = 158.

Robert Israel, Table of n, a(n) for n = 1..302
Wikipedia, Toeplitz Matrix

Cf. A005843, A000040, A007504, A306457, A306192.

f:= proc(n) uses LinearAlgebra;
Determinant(ToeplitzMatrix([seq(ithprime(i),i=2*n-1..n+1,-1),seq(ithprime(i),i=1..n)]))
end proc:
map(f, [$1..20]); # Robert Israel, Aug 30 2018

p[i_]:=Prime[i]; a[n_]:=Det[ToeplitzMatrix[Join[{p[1]},Array[p,n-1,{n+1,2*n-1}]],Array[p,n]]]; Array[a,20]

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, prime(j), if (j==1, prime(n+i-1))))); for (i=2, n, for (j=2, n, m[i,j] = m[i-1, j-1];);); m;}
a(n) = matdet(tm(n)); \\ Michel Marcus, Mar 17 2019

A306457 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n) and whose first column consists of prime(1), prime(n + 1), ..., prime(2*n - 1).

Original entry on oeis.org

1, 2, 19, 546, 40851, 4747510, 986799301, 292666754602, 135134321711681, 80312872924339660, 55242523096584443271, 52058868505260739019880, 55579419798019716586180451, 72402676504369062268839297084, 120521257466525185305708420453019, 234000358527930078723939842673115488

Offset: 0

Stefano Spezia, Feb 17 2019

nonn

The trace of the matrix M(n) is A005843(n).
The sum of the first row of the matrix M(n) is A007504(n).
The determinant of the matrix M(n) is A318173(n).
For n > 1, the subdiagonal sum of the matrix M(n) is A306192(n).

			For n = 1 the matrix M(1) is
    2
with permanent a(1) = 2.
For n = 2 the matrix M(2) is
    2, 3
    5, 2
with permanent a(2) = 19.
For n = 3 the matrix M(3) is
    2, 3, 5
    7, 2, 3
   11, 7, 2
with permanent a(3) = 546.

Stefano Spezia, Table of n, a(n) for n = 0..35
Wikipedia, Toeplitz Matrix

Cf. A005843, A000040, A007504, A318173, A306192.

f:= proc(n) uses LinearAlgebra; `if`(n=0, 1, Permanent(ToeplitzMatrix([seq(ithprime(i), i=2*n-1..n+1, -1), seq(ithprime(i), i=1..n)]))) end proc: map(f, [$0..15]);

p[i_]:=Prime[i];a[n_]:=If[n==0,1,Permanent[ToeplitzMatrix[Join[{p[1]},Array[p,n-1,{n+1,2*n-1}]],Array[p,n]]]];Array[a,15,0]

tm(n) = {my(m = matrix(n, n, i, j, if (i==1, prime(j), if (j==1, prime(n+i-1))))); for (i=2, n, for (j=2, n, m[i,j] = m[i-1, j-1];);); m;}
a(n) = matpermanent(tm(n)); \\ Michel Marcus, Mar 16 2019

a(0) = 1 prepended by Stefano Spezia, Dec 06 2019

A340781 a(n) = (n - 1)*prime(n + 1) mod prime(n).

Original entry on oeis.org

0, 2, 4, 5, 8, 7, 12, 9, 2, 18, 29, 7, 24, 9, 37, 37, 32, 41, 5, 38, 47, 5, 49, 6, 96, 50, 1, 54, 3, 67, 120, 55, 64, 52, 68, 59, 59, 148, 61, 61, 80, 48, 84, 172, 88, 142, 130, 188, 96, 196, 67, 102, 38, 67, 67, 67, 112, 71, 232, 118, 34, 268, 248, 126, 256, 276

Offset: 1

Stefano Spezia, Jan 21 2021

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000040, A033286, A306192, A340128, A340649.

Table[Mod[(n-1)Prime[n+1],Prime[n]],{n,66}]

a(n) = ((n-1)*prime(n+1)) % prime(n); \\ Michel Marcus, Jan 21 2021

a(n) = A306192(n) mod A000040(n).

cp's OEIS Frontend

A318173 The determinant of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n) and whose first column consists of prime(1), prime(n + 1), ..., prime(2*n - 1).

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A306457 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n) and whose first column consists of prime(1), prime(n + 1), ..., prime(2*n - 1).

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A340781 a(n) = (n - 1)*prime(n + 1) mod prime(n).

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