A248579 a(n) = the smallest numbers k such that n*T(k)-1 and n*T(k)+1 are twin primes or 0 if no solution exists for n where T(k) = A000217(k) = k-th triangular number.

3, 2, 3, 1, 3, 1, 3, 0, 0, 2, 12, 1, 11, 2, 4, 5, 3, 1, 12, 2, 24, 6, 3, 2, 3, 12, 4, 5, 20, 1, 27, 3, 3, 2, 11, 2, 56, 3, 7, 3, 32, 1, 44, 5, 3, 2, 3, 11, 12, 2, 7, 3, 15, 5, 20, 14, 4, 3, 32, 1, 27, 6, 8, 2, 8, 2, 11, 5, 7, 3, 167, 1, 20, 9, 12, 2, 3, 18

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.

a(n) > 0 for 10 <= n <= 100000. - Robert Israel, Aug 10 2023

Examples

			a(5) = 3 because 3 is the smallest number k with this property: 5*T(3) -/+ 1 = 5*6 -+ 1 = 29 and 31 (twin primes).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A016754, A060544, A248580.

a(n) = 1: A014574.

Programs

Magma

A248579:=func; [A248579(n): n in[1..100]]

Maple

f:= proc(n) local k;
   for k from 1 do if isprime(n*k*(k+1)/2+1) and isprime(n*k*(k+1)/2-1) then return k fi od:
end proc;
f(8):= 8: f(9):= 0:
map(f, [$1..100]); # Robert Israel, Aug 10 2023

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;} \\ Michel Marcus, Nov 05 2014