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A257833 Table T(k, n) of smallest bases b > 1 such that p = prime(n) satisfies b^(p-1) == 1 (mod p^k), read by antidiagonals.

%I A257833 #48 May 18 2022 09:37:59
%S A257833 5,8,9,7,26,17,18,57,80,33,3,18,182,242,65,19,124,1047,1068,728,129,
%T A257833 38,239,1963,1353,1068,2186,257,28,158,239,27216,34967,32318,6560,513,
%U A257833 28,333,4260,109193,284995,82681,110443,19682,1025,14,42,2819,15541,861642,758546,2387947,280182,59048,2049
%N A257833 Table T(k, n) of smallest bases b > 1 such that p = prime(n) satisfies b^(p-1) == 1 (mod p^k), read by antidiagonals.
%H A257833 Chai Wah Wu, <a href="/A257833/b257833.txt">Table of n, a(n) for n = 2..10000</a>
%e A257833 T(3, 5) = 124, since prime(5) = 11 and the smallest b such that b^10 == 1 (mod 11^3) is 124.
%e A257833 Table starts
%e A257833   k\n|    1     2       3        4       5       6         7
%e A257833   ---+----------------------------------------------------------
%e A257833    2 |    5     8       7       18       3      19        38 ...
%e A257833    3 |    9    26      57       18     124     239       158 ...
%e A257833    4 |   17    80     182     1047    1963     239      4260 ...
%e A257833    5 |   33   242    1068     1353   27216  109193     15541 ...
%e A257833    6 |   65   728    1068    34967  284995  861642    390112 ...
%e A257833    7 |  129  2186   32318    82681  758546 6826318  21444846 ...
%e A257833    8 |  257  6560  110443  2387947 9236508 6826318 112184244 ...
%e A257833    9 |  513 19682  280182 14906455 ....
%e A257833   10 | 1025 59048 3626068 ....
%e A257833   ...
%o A257833 (PARI) for(k=2, 10, forprime(p=2, 25, b=2; while(Mod(b, p^k)^(p-1)!=1, b++); print1(b, ", ")); print(""))
%o A257833 (PARI) T(k,n) = my(p=prime(n), v=List([2])); if(n==1, return(2^k+1)); for(i=1, k, w=List([]); for(j=1, #v, forstep(b=v[j], p^i-1, p^(i-1), if(Mod(b, p^i)^p==b, listput(w, b)))); v=Vec(w)); vecmin(v); \\ _Jinyuan Wang_, May 17 2022
%o A257833 (Python)
%o A257833 from itertools import count, islice
%o A257833 from sympy import prime
%o A257833 from sympy.ntheory.residue_ntheory import nthroot_mod
%o A257833 def A257833_T(n,k): return 2**k+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**k,True)[1])
%o A257833 def A257833_gen(): # generator of terms
%o A257833     yield from (A257833_T(n,i-n+2) for i in count(1) for n in range(i,0,-1))
%o A257833 A257833_list = list(islice(A257833_gen(),50)) # _Chai Wah Wu_, May 17 2022
%Y A257833 Column 1 of table is A000051.
%Y A257833 Column 2 of table is A024023 (with offset 2).
%Y A257833 Column 3 of table is A034939 (with offset 2).
%Y A257833 Rows k=2-10 give: A039678, A249275, A353937, A353938, A353939, A353940, A353941, A353942, A353943.
%K A257833 nonn,tabl
%O A257833 2,1
%A A257833 _Felix Fröhlich_, May 10 2015