A248580 a(n) = the smallest triangular number T(k) such that n*T(k)-1 and n*T(k)+1 are twin primes or 0 if no solution exists for n; T(k) = A000217(k) = k-th triangular number.

6, 3, 6, 1, 6, 1, 6, 0, 0, 3, 78, 1, 66, 3, 10, 15, 6, 1, 78, 3, 300, 21, 6, 3, 6, 78, 10, 15, 210, 1, 378, 6, 6, 3, 66, 3, 1596, 6, 28, 6, 528, 1, 990, 15, 6, 3, 6, 66, 78, 3, 28, 6, 120, 15, 210, 105, 10, 6, 528, 1, 378, 21, 36, 3, 36, 3, 66, 15, 28, 6

Offset: 1

Author

Jaroslav Krizek, Oct 25 2014

Keywords

• nonn

Comments

For n = 8 and 9 there are no triangular numbers T(k) such that n*T(k)-+1 are twin primes.

a(8) = 0 because 8*T(k)+1 = A016754(k) = composite number for k >= 1.

a(9) = 0 because 9*T(k)+1 = A060544(k+1) = composite number for k >= 1.

Are there numbers n > 9 such that a(n) = 0? If a(n) = 0 for n > 9, n must be bigger than 4000.

Examples

			a(5) = 6 because 6 is the smallest smallest triangular number with this property: 5*6 -+ 1 = 29 and 31 (twin primes).
		

Crossrefs

Cf. A000217, A016754, A060544, A248579.

Programs

Magma

A248580:=func; [A248580(n): n in[1..100]]

Mathematica

a248580[n_Integer] := Catch@Module[{T, k}, T[i_] := i (i + 1)/2; Do[If[And[PrimeQ[n*T[k] + 1], PrimeQ[n*T[k] - 1]], Throw[T[k]], 0], {k, 1, 10^4}] /. Null -> 0]; a248580 /@ Range[70] (* Michael De Vlieger, Nov 12 2014 *)

PARI

a(n) = {if ((n==8) || (n==9), return (0)); k = 1; while (!isprime(n*k*(k+1)/2-1) || !isprime(n*k*(k+1)/2+1), k++); k*(k+1)/2; } \\ Michel Marcus, Nov 12 2014

Formula

a(n) = A000217(A248579(n)).