A353937 Smallest b > 1 such that b^(p-1) == 1 (mod p^4) for p = prime(n).

17, 80, 182, 1047, 1963, 239, 4260, 2819, 19214, 2463, 15714, 51344, 20677, 3038, 224444, 189323, 11550, 397575, 201305, 15384, 840838, 1372873, 1576656, 278454, 1721322, 48072, 281007, 119551, 252595, 1001934, 3489507, 2489004, 598987, 3082551, 6136759, 3928984

Offset: 1

Author

Felix Fröhlich, May 12 2022

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Row k = 4 of A257833.

Cf. similar sequences for p^k: A039678 (k=2), A249275 (k=3), A353938 (k=5), A353939 (k=6), A353940 (k=7), A353941 (k=8), A353942 (k=9), A353943 (k=10).

Programs

Maple

f:= proc(j) local p,b,i;
  p:= ithprime(j);
  b:= numtheory:-primroot(p^4) &^ (p^3) mod p^4;
  min(seq(b &^i mod p^4, i=1..p-2))
end proc:
f(1):= 17:
map(f, [$1..40]); # Robert Israel, Dec 19 2024

Mathematica

a[n_] := Module[{p = Prime[n], b = 2}, While[PowerMod[b, p - 1, p^4] != 1, b++]; b]; Array[a, 20] (* Amiram Eldar, May 12 2022 *)

PARI

a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^4)^(p-1)==1, return(b)))

Python

from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A353937(n): return 2**4+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**4,True)[1]) # Chai Wah Wu, May 17 2022