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A257833
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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.
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12
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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
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OFFSET
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2,1
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LINKS
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EXAMPLE
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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 ....
...
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PROG
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(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))
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CROSSREFS
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Column 2 of table is A024023 (with offset 2).
Column 3 of table is A034939 (with offset 2).
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KEYWORD
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AUTHOR
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STATUS
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approved
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