A256517 Let c be the n-th composite number. Then a(n) is the smallest base b > 1 such that b^(c-1) == 1 (mod c^2), i.e., such that c is a 'Wieferich pseudoprime'.

17, 37, 65, 80, 101, 145, 197, 26, 257, 325, 401, 197, 485, 577, 182, 677, 728, 177, 901, 1025, 485, 1157, 99, 1297, 1445, 170, 1601, 1765, 1937, 82, 2117, 2305, 1047, 2501, 577, 529, 2917, 1451, 3137, 721, 3365, 3601, 3845, 244, 4097, 99, 1945, 4625, 530

Offset: 1

Author

Felix Fröhlich, Apr 01 2015

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..3000 from Felix Fröhlich)

Crossrefs

Cf. A039678, A242742, A244752, A255885.

Cf. A185103, A002808.

Programs

Mathematica

c = Select[Range@ 69, CompositeQ]; f[c_] := Block[{b = 2}, While[Mod[b^(c - 1), c^2] != 1, b++]; b]; f /@ c (* Michael De Vlieger, Apr 03 2015 *)

PARI

forcomposite(c=1, 1e3, b=2; while(Mod(b, c^2)^(c-1)!=1, b++); print1(b, ", "))

Python

from sympy import composite
from sympy.ntheory.residue_ntheory import nthroot_mod
def A256517(n):
    z = nthroot_mod(1,(c := composite(n))-1,c**2,True)
    return int(z[0]+c**2 if len(z) == 1 else z[1]) # Chai Wah Wu, May 18 2022

Formula

a(n) = A185103(A002808(n)-1). - Bill McEachen, Nov 27 2021