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.

5, 8, 9, 7, 26, 17, 18, 57, 80, 33, 3, 18, 182, 242, 65, 19, 124, 1047, 1068, 728, 129, 38, 239, 1963, 1353, 1068, 2186, 257, 28, 158, 239, 27216, 34967, 32318, 6560, 513, 28, 333, 4260, 109193, 284995, 82681, 110443, 19682, 1025, 14, 42, 2819, 15541, 861642, 758546, 2387947, 280182, 59048, 2049

Offset

2,1

Links

Chai Wah Wu, Table of n, a(n) for n = 2..10000

Example

T(3, 5) = 124, since prime(5) = 11 and the smallest b such that b^10 == 1 (mod 11^3) is 124.
Table starts
  k\n|    1     2       3        4       5       6         7
  ---+----------------------------------------------------------
   2 |    5     8       7       18       3      19        38 ...
   3 |    9    26      57       18     124     239       158 ...
   4 |   17    80     182     1047    1963     239      4260 ...
   5 |   33   242    1068     1353   27216  109193     15541 ...
   6 |   65   728    1068    34967  284995  861642    390112 ...
   7 |  129  2186   32318    82681  758546 6826318  21444846 ...
   8 |  257  6560  110443  2387947 9236508 6826318 112184244 ...
   9 |  513 19682  280182 14906455 ....
  10 | 1025 59048 3626068 ....
  ...

Prog

(PARI) for(k=2, 10, forprime(p=2, 25, b=2; while(Mod(b, p^k)^(p-1)!=1, b++); print1(b, ", ")); print(""))
(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
(Python)
from itertools import count, islice
from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
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])
def A257833_gen(): # generator of terms
    yield from (A257833_T(n, i-n+2) for i in count(1) for n in range(i, 0, -1))
A257833_list = list(islice(A257833_gen(), 50)) # Chai Wah Wu, May 17 2022

Crossrefs

Column 1 of table is A000051.

Column 2 of table is A024023 (with offset 2).

Column 3 of table is A034939 (with offset 2).

Rows k=2-10 give: A039678, A249275, A353937, A353938, A353939, A353940, A353941, A353942, A353943.

Sequence in context: A336003 A314578 A314577 * A021633 A178309 A259071

Adjacent sequences: A257830 A257831 A257832 * A257834 A257835 A257836

Keyword

nonn,tabl

Author

Felix Fröhlich, May 10 2015

Status

approved