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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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Map[Mod[((# - 1)/2)!, #] &, Select[4 Range[192] + 1, PrimeQ]] (* Michael De Vlieger, Oct 15 2020 *)
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PROG
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(PARI) my(v=select(p->p%4==1, primes(100))); apply(x->(((x-1)/2)! % x), v) \\ Michel Marcus, Aug 07 2020
(Python) 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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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