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
Examples
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 .... ...
Links
- Chai Wah Wu, Table of n, a(n) for n = 2..10000
Crossrefs
Programs
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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
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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