A249275 a(n) is the smallest b > 1 such that p = prime(n) satisfies b^(p-1) == 1 (mod p^3).

9, 26, 57, 18, 124, 239, 158, 333, 42, 1215, 513, 691, 1172, 3038, 295, 1468, 2511, 15458, 3859, 6372, 923, 1523, 5436, 1148, 412, 4943, 4432, 5573, 476, 68, 21304, 30422, 6021, 8881, 33731, 25667, 3868, 3170, 17987, 26626, 43588, 7296, 14628, 22076, 138057

Offset: 1

Author

Felix Fröhlich, Oct 24 2014

Keywords

• nonn

Comments

a(n) >= A039678(n) for all n.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A039678, A242742.

Programs

Mathematica

Array[Block[{b = 2}, While[PowerMod[b, # - 1, #^3] != 1, b++]; b] &@ Prime@ # &, 45] (* Michael De Vlieger, Nov 25 2018 *)
dpa[n_]:=Module[{p=Prime[n], a=9}, While[PowerMod[a, p - 1, p^3]!=1, a++]; a]; Array[dpa, 50] (* Vincenzo Librandi, Nov 30 2018 *)

PARI

a(n) = my(p=prime(n)); for(b=2, oo, if(Mod(b, p^3)^(p-1)==1, return(b)))

Python

from sympy import prime
def a(n):
    b, p = 2, prime(n)
    p3 = p**3
    while pow(b, p-1, p3) != 1: b += 1
    return b
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Sep 26 2021

Python

from sympy import prime
from sympy.ntheory.residue_ntheory import nthroot_mod
def A249275(n): return 2**3+1 if n == 1 else int(nthroot_mod(1,(p:= prime(n))-1,p**3,True)[1]) # Chai Wah Wu, May 17 2022

Extensions

Edited by Felix Fröhlich, Nov 24 2018