A330406 a(n) is the smallest prime q such that q^((p-1)/2) == -1 (mod p), where p = A002144(n) is the n-th prime congruent to 1 mod 4.

2, 2, 3, 2, 2, 3, 2, 2, 5, 3, 5, 2, 2, 3, 3, 2, 2, 2, 2, 5, 2, 2, 3, 7, 3, 2, 2, 3, 2, 5, 2, 5, 2, 3, 2, 2, 2, 3, 7, 2, 5, 3, 5, 2, 2, 3, 2, 2, 3, 5, 3, 7, 2, 3, 3, 2, 2, 5, 2, 2, 2, 2, 2, 3, 7, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 5, 2, 3, 3, 2, 11, 2, 2, 5, 3, 2, 2, 2, 3, 2, 2, 11, 5, 2, 3, 11, 2, 3, 2, 2, 7, 2, 3, 5, 2, 7, 3, 2, 2

Offset

1,1

Comments

Subset of A053760 corresponding to p == 1 (mod 4).

A002144(n) = p is a sum of two integer squares (Fermat): p = a^2 + b^2. To find a and b, calculate gcd(p, A330406(n)^((p-1)/4)+i) = a + bi in the Gaussian integers.

Links

Table of n, a(n) for n=1..109.

Example

Let p = A002144(30)= 313. Then (p-1)/2 = 156. Now 2^156 == 3^156 == 1 (mod p) but 5^156 == -1 (mod p).  Thus A330406(30)=5.

Mathematica

Map[Block[{q = 2}, While[PowerMod[q, (# - 1)/2, #] != # - 1, q = NextPrime@ q]; q] &, Select[4 Range[350] + 1, PrimeQ]] (* Michael De Vlieger, Dec 29 2019 *)

Prog

(PARI) A002144 = select(p->p%4==1, primes(2200));
A330406 = vector(1000); for(i=1, 1000, my(p=A002144[i]); forprime(j=1, 20, my(x=Mod(j, p)^((p-1)/2)); if(x+1, , A330406[i]=j; break)))
A330406

Crossrefs

Cf. A002144, A053760.

Sequence in context: A046027 A283671 A046028 * A125954 A122443 A262945

Adjacent sequences: A330403 A330404 A330405 * A330407 A330408 A330409

Keyword

nonn

Author

Nicholas C. Singer, Dec 13 2019

Status

approved