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

%I A353937 #17 Dec 19 2024 06:57:11
%S A353937 17,80,182,1047,1963,239,4260,2819,19214,2463,15714,51344,20677,3038,
%T A353937 224444,189323,11550,397575,201305,15384,840838,1372873,1576656,
%U A353937 278454,1721322,48072,281007,119551,252595,1001934,3489507,2489004,598987,3082551,6136759,3928984
%N A353937 Smallest b > 1 such that b^(p-1) == 1 (mod p^4) for p = prime(n).
%H A353937 Robert Israel, <a href="/A353937/b353937.txt">Table of n, a(n) for n = 1..10000</a>
%p A353937 f:= proc(j) local p,b,i;
%p A353937   p:= ithprime(j);
%p A353937   b:= numtheory:-primroot(p^4) &^ (p^3) mod p^4;
%p A353937   min(seq(b &^i mod p^4, i=1..p-2))
%p A353937 end proc:
%p A353937 f(1):= 17:
%p A353937 map(f, [$1..40]); # _Robert Israel_, Dec 19 2024
%t A353937 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 *)
%o A353937 (PARI) a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^4)^(p-1)==1, return(b)))
%o A353937 (Python)
%o A353937 from sympy import prime
%o A353937 from sympy.ntheory.residue_ntheory import nthroot_mod
%o A353937 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
%Y A353937 Row k = 4 of A257833.
%Y A353937 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).
%K A353937 nonn
%O A353937 1,1
%A A353937 _Felix Fröhlich_, May 12 2022