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

3, 8, 4, 17, 6, 32, 30, 50, 46, 55, 75, 10, 76, 98, 100, 105, 28, 93, 19, 112, 14, 107, 89, 177, 241, 82, 60, 228, 155, 25, 203, 148, 136, 311, 269, 115, 334, 20, 143, 392, 179, 67, 109, 413, 208, 235, 52, 118, 86, 553, 516, 476, 35, 194, 154, 504, 106, 58, 26, 566, 613, 353, 670, 722

Offset

1,1

Comments

For more information see A336883.

Links

Hiroyuki Hara, Table of n, a(n) for n = 1..4783 [reformatted and restored by Georg Fischer, Oct 16 2020]

Formula

a(n) == (A002144(n) - 2)!/((A002144(n) - 1)/2)! == -((A002144(n) - 1)/2)! == -A336883(n) == A002144(n) - A336883(n) mod A002144(n).

Example

p(1)=5: (5-2)!=2*3=A336883(1)*a(1)==1 mod 5. 5=2+3.
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.
a(n)=4: A336883(n)=(k*4+1)/(4-k)=(3*4+1)/(4-3)=13, k=3. p(n)=13+4=17.
a(n)=17: A336883(n)=(k*17+1)/(17-k)=(7*17+1)/(17-7)=12, k=7. p(n)=12+17=29.

Mathematica

v = Select[Prime[Range[1000]], Mod[#, 4] == 1&];
v - Mod[((v-1)/2)!, v] (* Jean-François Alcover, Oct 24 2020, after PARI *)

Prog

(PARI) my(v=select(p->p%4==1, primes(100))); apply(x->x - (((x-1)/2)! % x), v) \\ Michel Marcus, Aug 07 2020
(Python) n_start=5
n_end=n_start+100000
k=1
for n in range(n_start, n_end, 4):
    c=(n-1)//2
    r=1
    for i in range(2, c+1):
        r=r*i % n
        if r==0:
            break
    if (n-r)*r % n ==1:
        print(k, n-r)
        k = k + 1
# modified by Georg Fischer, Oct 16 2020

Crossrefs

Cf. A336883, A002144 (Pythagorean primes), A206549, A209874, A256011, A186814, A282538.

Sequence in context: A127438 A303217 A106292 * A072247 A051359 A322470

Adjacent sequences: A336881 A336882 A336883 * A336885 A336886 A336887

Keyword

nonn

Author

Hiroyuki Hara, Aug 06 2020

Status

approved