A336884 -id:A336884 - cp's OEIS frontend

A336883 a(n) = ((A002144(n) - 1)/2)! (mod A002144(n)) where A002144(n) is the n-th Pythagorean prime.

2, 5, 13, 12, 31, 9, 23, 11, 27, 34, 22, 91, 33, 15, 37, 44, 129, 80, 162, 81, 183, 122, 144, 64, 16, 187, 217, 53, 138, 288, 114, 189, 213, 42, 104, 274, 63, 381, 266, 29, 254, 382, 348, 48, 301, 286, 489, 439, 483, 24, 77, 125, 578, 423, 487, 149, 555, 615, 651, 135, 96, 380, 87, 39, 707

Offset: 1

Hiroyuki Hara, Aug 06 2020

nonn

Let p(n) = A002144(n) be the n-th Pythagorean prime.
Pythagorean prime p can be divided into a pair of integers (a,b) such as p =a+b and a*b==1 mod p. And (p-2)!==1 mod p because of Wilson's Theorem (p-1)!==-1 mod p. It can be divided into two parts (a,b) such as {2*3*4*...*((p(n)-1)/2)==a(n) mod p(n)} and {((p(n)-1)/2+1)*...*(p(n)-4)*(p(n)-3)*(p(n)-2)==-a(n)==(p(n)-a(n)) mod p(n)}. The pair numbers make a(n)+(p(n)-a(n))=p(n) and a(n)*(p(n)-a(n))==1 mod p(n). The left integer of the pair numbers is a(n). The right integer (p(n)-a(n)) is A336884(n).
The set of selecting odd numbers from {a(n)} and A336884 is A206549. The set of selecting even numbers from {a(n)} and A336884 is A209874 except for the number 1. A256011 never appears in {a(n)} or A336884. It is related to nonexistence of numbers that the largest prime factor of n^2+1 is greater than n.
The odd number of the difference |a(n)-A336884(n)|=|a(n)-(p(n)-a(n))|=|2*a(n)-p(n)| is A186814(n). A282538 never appears in the set of the difference |a(n)-A336884(n)|.
If p(n) is unknown, p(n) can be derived from a(n) using following equation. From a*b==1 mod p, a*b=k*p+1. With p=a+b, it can transform to b(n)=(k*a(n)+1)/(a(n)-k), k is an odd integer parameter when the fraction makes an integer. If there are many k's, select the minimum k in those. Then a(n)+b(n)=p(n). b(n) is A336884(n).

			p(1)=5: (5-2)!=2*3=a(1)*(5-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==a(2)*(13-a(2))==1 mod 13. 13=5+8.
a(n)=13: b(n)=(k*13+1)/(13-k)=(3*13+1)/(13-3)=4, k=3. p(n)=13+4=17.
a(n)=12: b(n)=(k*12+1)/(12-k)=(7*12+1)/(12-7)=17, k=7. p(n)=12+17=29.

Hiroyuki Hara, Table of n, a(n) for n = 1..4783 [reformatted and restored by _Georg Fischer_, Oct 15 2020]

Cf. A336884, A002144 (Pythagorean primes), A206549, A209874, A256011, A186814, A282538.

Map[Mod[((# - 1)/2)!, #] &, Select[4 Range[192] + 1, PrimeQ]] (* Michael De Vlieger, Oct 15 2020 *)

my(v=select(p->p%4==1, primes(100))); apply(x->(((x-1)/2)! % x), v) \\ Michel Marcus, Aug 07 2020

n_start=5
n_end=n_start+10000
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, r)
        k = k + 1
# modified by Georg Fischer, Oct 16 2020

cp's OEIS Frontend

A336883 a(n) = ((A002144(n) - 1)/2)! (mod A002144(n)) where A002144(n) is the n-th Pythagorean prime.

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