A174962 -id:A174962 - cp's OEIS frontend

A174960 Smallest prime p such that p + n*(n+1)/2 is prime, or 0 if no such prime exists.

2, 2, 2, 5, 3, 2, 2, 3, 5, 2, 0, 5, 5, 0, 2, 7, 3, 0, 2, 3, 13, 2, 0, 5, 7, 0, 2, 5, 3, 0, 2, 3, 13, 2, 0, 11, 7, 0, 2, 7, 3, 2, 0, 7, 7, 0, 0, 23, 5, 0, 2, 41, 3, 2, 2, 3, 5, 0, 0, 7, 17, 0, 0, 11, 3, 0, 2, 3, 5, 2, 0, 23, 5, 0, 2, 7, 13, 0, 2, 3, 11, 2, 0, 5, 11, 0, 0, 5, 3, 2, 0, 31, 5, 2, 0, 7, 7, 0, 0

Offset: 0

Michel Lagneau, Apr 02 2010

nonn

n(n+1)/2 = A000217(n).
If n(n+1)/2 is odd, m+n(n+1)/2 can be prime only for m = 2, since otherwise m+n(n+1)/2 is divisible by 2. Hence a(n) = 0 if n(n+1)/2 is odd and 2+n(n+1)/2 is not prime.
For n > 0 also smallest m such that all eigenvalues of the n X n matrix M_m,n are prime, where M_m,n(j,k) = j for j <> k, M_m,n(j,k) = m+j for j = k.
The eigenvalues of M_m,n are m+n(n+1)/2, and m with the multiplicity n-1; cf. reference for proof. Thus all eigenvalues can be prime only if m is prime.

			(in Maple notation)
For n = 1 and m = 2, eigenvals(matrix(1,1, [[3]])) = {3}, so a(1) = 2.
For n = 2 and m = 2, eigenvals(matrix(2,2, [[3,1],[2,4]]) = {2,5} so a(2) = 2.
For n = 3 and m = 2, eigenvals(matrix(3,3, [[3,1,1],[2,4,2],[3,3,5]])) = {2,2,8} and 8 is not prime; for m = 3, eigenvals(matrix(3,3, [[4,1,1],[2,5,2],[3,3,6]])) = {3,3,9} and 9 is not prime; for m = 5, eigenvals(matrix(3,3, [[6,1,1],[2,7,2],[3,3,8]])) = {5,5,11} and 11 is prime, so a(3) = 5;

J.-M. Monier, Algebre et geometrie, exercices corriges. Dunod, 1997, p. 78.

K. Brockhaus, Table of n, a(n) for n = 0..10000

Cf. A000217 (triangular numbers), A174962.

SmallestP:=function(n) for p in PrimesUpTo(1000) do if IsPrime(p + n*(n+1) div 2) then return p; end if; end for; return 0; end function; [SmallestP(n): n in [0..100]]; // Klaus Brockhaus, Apr 10 2010

SmallestQ:=function(n) for m in PrimesUpTo(1000) do E:=Eigenvalues(Matrix([&cat[ [j ne k select j else m+j]: k in [1..n]]: j in [1..n] ])); if forall(t){x: x in E | IsPrime(x[1])} then return m; end if; end for; return 0; end function; [2] cat [SmallestQ(n): n in [1..100]]; // Klaus Brockhaus, Apr 10 2010

with(numtheory):for n from 1 to 200 do:nn:=1:for k from 2 to 1000 do: x:=k + n*(n+1)/2:if (type(x,prime)=true)and(type(k,prime)=true)and nn=1 then print(k):nn:=2:else fi:od:od:

a[n_] := (p = 2; q = n*(n+1)/2; While[p > 0, If[ PrimeQ[p+q], Break[], p = If[ OddQ[q], 0, NextPrime[p]]]]; p); Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 03 2011 *)

Edited and corrected by Klaus Brockhaus, Apr 10 2010

A174963 Determinant of the symmetric n X n matrix M_n where M_n(j,k) = n for j = k, M_n(j,n) = n-j, M_n(n,k) = n-k, M_n(j,k) = 0 otherwise.

Original entry on oeis.org

1, 3, 12, 32, -625, -24624, -705894, -19922944, -588305187, -18500000000, -622498190424, -22414085849088, -862029149531797, -35320307409809408, -1537494104003906250, -70904672533321089024, -3454944623172347662151, -177423154932124201844736

Offset: 1

Michel Lagneau, Apr 02 2010

sign

			a(5) = det(M_5) = -625 where M_5 is the matrix
  [5 0 0 0 4]
  [0 5 0 0 3]
  [0 0 5 0 2]
  [0 0 0 5 1]
  [4 3 2 1 5]

J.-M. Monier, Algèbre et géometrie, exercices corrigés. Dunod, 1997, p. 78.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A174962.

[ n^n -((n-1)*n*(2*n-1)/6)*n^(n-2): n in [1..18] ]; // Klaus Brockhaus, Apr 11 2010

[ Determinant( SymmetricMatrix( &cat[ [ i lt j select 0 else n: i in [1..j] ]: j in [1..n-1] ] cat [ 1+((n-1-k) mod n): k in [1..n] ] ) ): n in [1..18] ]; // Klaus Brockhaus, Apr 11 2010

with(numtheory):for n from 1 to 25 do:x:=n^n -((n-1)*n*(2*n-1)/6)*n^(n-2):print(x):od:

M[j_,k_,n_]:=If[j==k,n,If[k==n,n-j,If[j==n,n-k,0]]]; a[n_]:=Det[Table[M[i,j,n],{i,n},{j,n}]]; Array[a,18] (* Stefano Spezia, Aug 11 2025 *)

a(n) = n^n - ((n-1)*n*(2*n-1)/6)*n^(n-2).

Edited by Klaus Brockhaus, Apr 11 2010

A386975 a(n) is the permanent of the n X n matrix M_n with M_n(j,k) = j for j <> k, M_n(j,k) = n+j for j = k.

Original entry on oeis.org

1, 2, 14, 183, 3792, 114780, 4807728, 267380071, 19098388480, 1705287529422, 186174804704000, 24402257980061599, 3781731531452940288, 684046276855242721368, 142823583210894978115584, 34092816821609506532859375, 9226267072346511233190461440, 2809774286001810901571097532538

Offset: 0

Stefano Spezia, Aug 11 2025

nonn

			a(5) = permanent(M_5) = 114780 where M_5 is the matrix
  [6, 1, 1, 1,  1]
  [2, 7, 2, 2,  2]
  [3, 3, 8, 3,  3]
  [4, 4, 4, 9,  4]
  [5, 5, 5, 5, 10]

Cf. A174962 (determinants), A386974.

M[j_,k_,n_]:=If[j!=k,j,If[j==k,n+j]]; a[n_]:=Permanent[Table[M[i,j,n],{i,n},{j,n}]];Join[{1}, Array[a,17]]

a(n) = matpermanent(matrix(n, n, j, k, if (j==k, n+j, j))); \\ Michel Marcus, Aug 12 2025

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A174960 Smallest prime p such that p + n*(n+1)/2 is prime, or 0 if no such prime exists.

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A174963 Determinant of the symmetric n X n matrix M_n where M_n(j,k) = n for j = k, M_n(j,n) = n-j, M_n(n,k) = n-k, M_n(j,k) = 0 otherwise.

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A386975 a(n) is the permanent of the n X n matrix M_n with M_n(j,k) = j for j <> k, M_n(j,k) = n+j for j = k.

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