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A336884 a(n) = A002144(n) - A336883(n) where A002144(n) is the n-th Pythagorean prime.

%I A336884 #53 Oct 24 2020 17:10:49
%S A336884 3,8,4,17,6,32,30,50,46,55,75,10,76,98,100,105,28,93,19,112,14,107,89,
%T A336884 177,241,82,60,228,155,25,203,148,136,311,269,115,334,20,143,392,179,
%U A336884 67,109,413,208,235,52,118,86,553,516,476,35,194,154,504,106,58,26,566,613,353,670,722
%N A336884 a(n) = A002144(n) - A336883(n) where A002144(n) is the n-th Pythagorean prime.
%C A336884 For more information see A336883.
%H A336884 Hiroyuki Hara, <a href="/A336884/b336884.txt">Table of n, a(n) for n = 1..4783</a> [reformatted and restored by _Georg Fischer_, Oct 16 2020]
%F A336884 a(n) == (A002144(n) - 2)!/((A002144(n) - 1)/2)! == -((A002144(n) - 1)/2)! == -A336883(n) == A002144(n) - A336883(n) mod A002144(n).
%e A336884 p(1)=5: (5-2)!=2*3=A336883(1)*a(1)==1 mod 5. 5=2+3.
%e A336884 p(2)=13: (13-2)!=(2*3*4*5*6)*(7*8*9*10*11)=(2*3*4*5*6)*((p-6)*(p-5)*(p-4)*(p-3)*(p-2))==5*(-5)==5*(13-5)=5*8==A336883(2)*a(2)==1 mod 13. 13=5+8.
%e A336884 a(n)=4: A336883(n)=(k*4+1)/(4-k)=(3*4+1)/(4-3)=13, k=3. p(n)=13+4=17.
%e A336884 a(n)=17: A336883(n)=(k*17+1)/(17-k)=(7*17+1)/(17-7)=12, k=7. p(n)=12+17=29.
%t A336884 v = Select[Prime[Range[1000]], Mod[#, 4] == 1&];
%t A336884 v - Mod[((v-1)/2)!, v] (* _Jean-François Alcover_, Oct 24 2020, after PARI *)
%o A336884 (PARI) my(v=select(p->p%4==1, primes(100))); apply(x->x - (((x-1)/2)! % x), v) \\ _Michel Marcus_, Aug 07 2020
%o A336884 (Python) n_start=5
%o A336884 n_end=n_start+100000
%o A336884 k=1
%o A336884 for n in range(n_start, n_end, 4):
%o A336884     c=(n-1)//2
%o A336884     r=1
%o A336884     for i in range(2, c+1):
%o A336884         r=r*i % n
%o A336884         if r==0:
%o A336884             break
%o A336884     if (n-r)*r % n ==1:
%o A336884         print(k, n-r)
%o A336884         k = k + 1
%o A336884 # modified by Georg Fischer, Oct 16 2020
%Y A336884 Cf. A336883, A002144 (Pythagorean primes), A206549, A209874, A256011, A186814, A282538.
%K A336884 nonn
%O A336884 1,1
%A A336884 _Hiroyuki Hara_, Aug 06 2020