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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.
12
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
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).
Sequence in context: A336003 A314578 A314577 * A021633 A178309 A259071
KEYWORD
nonn,tabl
AUTHOR
Felix Fröhlich, May 10 2015
STATUS
approved