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

1,1

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.

Links

Table of n, a(n) for n=1..70.

Formula

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

Example

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

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 *)

Prog

(Magma) A248580:=func<n|exists(r){m*(m+1)/2:m in[1..1000000] | IsPrime(n*m*(m+1) div 2+1) and  IsPrime(n*m*(m+1) div 2-1)}select r else 0>; [A248580(n): n in[1..100]]
(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

Crossrefs

Cf. A000217, A016754, A060544, A248579.

Sequence in context: A019151 A367976 A143506 * A008567 A233700 A195436

Adjacent sequences: A248577 A248578 A248579 * A248581 A248582 A248583

Keyword

nonn

Author

Jaroslav Krizek, Oct 25 2014

Status

approved