A174962 -id:A174962

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A174960

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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.

REFERENCES

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

LINKS

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

EXAMPLE

(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;

MAPLE

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:

MATHEMATICA

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 *)

PROG

(Magma) 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

(Magma) 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

CROSSREFS

Cf. A000217 (triangular numbers), A174962.

KEYWORD

nonn

AUTHOR

Michel Lagneau, Apr 02 2010

EXTENSIONS

Edited and corrected by Klaus Brockhaus, Apr 10 2010

STATUS

approved

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.

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

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

LINKS

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

FORMULA

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

EXAMPLE

a(5) = determinant(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]

MAPLE

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:

PROG

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

(Magma) [ 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

CROSSREFS

Cf. A174962.