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
Keywords
Examples
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.
Links
- Hiroyuki Hara, Table of n, a(n) for n = 1..4783 [reformatted and restored by _Georg Fischer_, Oct 16 2020]
Programs
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Mathematica
v = Select[Prime[Range[1000]], Mod[#, 4] == 1&]; v - Mod[((v-1)/2)!, v] (* Jean-François Alcover, Oct 24 2020, after PARI *)
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PARI
my(v=select(p->p%4==1, primes(100))); apply(x->x - (((x-1)/2)! % x), v) \\ Michel Marcus, Aug 07 2020
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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
Comments