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

Author

Michel Lagneau, Apr 02 2010

Keywords

• nonn

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.

Examples

			(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;
		

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

Crossrefs

Cf. A000217 (triangular numbers), A174962.

Programs

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

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 *)

Extensions

Edited and corrected by Klaus Brockhaus, Apr 10 2010