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Smallest b > 1 such that b^(p-1) == 1 (mod p^5) for p = prime(n).
8

%I #11 May 18 2022 07:55:05

%S 33,242,1068,1353,27216,109193,15541,133140,495081,1115402,2754849,

%T 1353359,649828,3228564,2359835,4694824,7044514,28538377,1111415,

%U 77588426,16178110,2553319,9571390,158485540,18664438,146773512,45639527,448251412,48834112,141076650

%N Smallest b > 1 such that b^(p-1) == 1 (mod p^5) for p = prime(n).

%t a[n_] := Module[{p = Prime[n], b = 2}, While[PowerMod[b, p - 1, p^5] != 1, b++]; b]; Array[a, 12] (* _Amiram Eldar_, May 12 2022 *)

%o (PARI) a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^5)^(p-1)==1, return(b)))

%o (Python)

%o from sympy import prime

%o from sympy.ntheory.residue_ntheory import nthroot_mod

%o def A353938(n): return 2**5+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**5,True)[1]) # _Chai Wah Wu_, May 17 2022

%Y Row k = 5 of A257833.

%Y Cf. similar sequences for p^k: A039678 (k=2), A249275 (k=3), A353937 (k=4), A353939 (k=6), A353940 (k=7), A353941 (k=8), A353942 (k=9), A353943 (k=10).

%K nonn

%O 1,1

%A _Felix Fröhlich_, May 12 2022