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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.
1
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.
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)))
CROSSREFS
Sequence in context: A046027 A283671 A046028 * A125954 A122443 A262945
KEYWORD
nonn
AUTHOR
Nicholas C. Singer, Dec 13 2019
STATUS
approved