A248580 - OEIS

%I #9 Sep 08 2022 08:46:10

%S 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,

%T 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,

%U 15,210,105,10,6,528,1,378,21,36,3,36,3,66,15,28,6

%N 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.

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

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

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

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

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

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

%t 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 *)

%o (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]]

%o (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

%Y Cf. A000217, A016754, A060544, A248579.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Oct 25 2014