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

1,1

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.

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

Links

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

Example

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).

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

Prog

(Magma) A248579:=func<n|exists(r){m: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>; [A248579(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; } \\ Michel Marcus, Nov 05 2014

Crossrefs

Cf. A000217, A016754, A060544, A248580.

a(n) = 1: A014574.

Sequence in context: A236228 A082391 A283977 * A296992 A304783 A316830

Adjacent sequences: A248576 A248577 A248578 * A248580 A248581 A248582

Keyword

nonn

Author

Jaroslav Krizek, Oct 25 2014

Status

approved