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
Keywords
Examples
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.
Programs
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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 *) -
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
Comments