A219313 Smallest number k such that LegendreP[2*n, k] is prime.

3, 7, 7, 3, 41, 5, 89, 23, 21, 35, 55, 5, 181, 511, 241, 83, 709, 401, 3653, 901, 137, 497, 1411, 121, 281, 209, 201, 191, 1667, 89, 39, 181, 233, 2783, 85, 911, 1717, 919, 97, 1163, 1319, 971, 361, 2371, 1573, 121, 817, 733, 1657, 1895, 509, 431, 2399, 1483

Offset: 1

Author

Michel Lagneau, Nov 17 2012

Keywords

• nonn

Comments

LegendreP [2*n, x] is the 2*n th Legendre polynomial of the first kind evaluated at x.

The corresponding primes are in A219315.

Examples

			a(1) = 3 because LegendreP[2*1, x] = (3x^2 - 1)/2 = P(x) and P(3) = 13 is prime.
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.

Links

• Robert Israel, Table of n, a(n) for n = 1..342
• Eric Weisstein's World of Mathematics, Legendre Polynomial

Crossrefs

Cf. A219315.

Programs

Maple

f:= proc(n) local p,k,x;
  p:= unapply(orthopoly[P](2*n,x),x);
  for k from 1 by 2 do if isprime(p(k)) then return k fi od
end proc:
map(f, [$1..60]); # Robert Israel, Dec 26 2024

Mathematica

Table[k = 0; While[!PrimeQ[LegendreP [2*n,k]], k++]; k, {n, 70}]

PARI

a(n)=my(P=pollegendre(2*n),k,t); while(denominator(t=subst(P,'x,k++))>1 || !ispseudoprime(t), ); k \\ Charles R Greathouse IV, Mar 18 2017