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 A209877 a(n) = A209874(n)/2: Least m > 0 such that 4*m^2 = -1 modulo the Pythagorean prime A002144(n). 2
 1, 4, 2, 6, 3, 16, 15, 25, 23, 17, 11, 5, 38, 49, 50, 22, 14, 40, 81, 56, 7, 61, 72, 32, 8, 41, 30, 114, 69, 144, 57, 74, 68, 21, 52, 137, 167, 10, 133, 196, 127, 191, 174, 24, 104, 143, 26, 59, 43, 12, 258, 238, 289, 97, 77, 252, 53, 29, 13, 283, 48, 190, 335, 361, 31, 228, 291, 159, 263, 123, 260, 325, 363, 247, 162 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also: Square root of -1/4 in Z/pZ, for Pythagorean primes p=A002144(n). Also: Least m>0 such that the Pythagorean prime p=A002144(n) divides 4(kp +/- m)^2+1 for all k>=0. In practice these can also be determined by searching the least N^2+1 whose least prime factor is p=A002144(n): For given p, all of these N will have a(n) or p-a(n) as remainder mod 2p. LINKS Robert Israel, Table of n, a(n) for n = 1..10000 EXAMPLE a(1)=1 since A002144(1)=5 and 4*1^2+1 is divisible by 5; as a consequence 4*(5k+/-1)^2+1 = 100k^2 +/- 40k + 5 is divisible by 5 for all k. a(2)=4 since A002144(2)=13 and 4*4^2+1 = 65 is divisible by 13, while 4*1^1+1=5, 4*2^2+1=17 and 4*3^2+1=37 are not. As a consequence, 4*(13k+/-4)^2+1 = 13(...)+4*4^1+1 is divisible by 13 for all k. MAPLE f:= proc(p) local m;    if not isprime(p) then return NULL fi;    m:= numtheory:-msqrt(-1/4, p);    min(m, p-m); end proc: map(f, [seq(i, i=5..1000, 4)]); # Robert Israel, Mar 13 2018 MATHEMATICA f[p_] := Module[{r}, r /. Solve[4 r^2 == -1, r, Modulus -> p] // Min]; f /@ Select[4 Range + 1, PrimeQ] (* Jean-François Alcover, Jul 27 2020 *) PROG (PARI) apply(p->lift(sqrt(Mod(-1, p)/4)), A002144) CROSSREFS Cf. A209874. Cf. A002496, A014442, A085722, A144255, A089120, A193432. Cf. A002314, A177979. Sequence in context: A275845 A281978 A325887 * A273667 A187109 A242599 Adjacent sequences:  A209874 A209875 A209876 * A209878 A209879 A209880 KEYWORD nonn AUTHOR M. F. Hasler, Mar 14 2012 STATUS approved

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Last modified August 12 23:19 EDT 2020. Contains 336440 sequences. (Running on oeis4.)