A214151 Numbers k from the set == 5 (mod 6) with the property that 3^((3*k-1)/2) == 3 (mod k) and 2^((k-1)/2) == (k-1) (mod k).

11, 59, 83, 107, 131, 179, 227, 251, 347, 419, 443, 467, 491, 563, 587, 659, 683, 827, 947, 971, 1019, 1091, 1163, 1187, 1259, 1283, 1307, 1427, 1451, 1499, 1523, 1571, 1619, 1667, 1787, 1811, 1907, 1931, 1979, 2003, 2027, 2099, 2243, 2267

Offset: 1

Author

Alzhekeyev Ascar M, Jul 05 2012

Keywords

• nonn

Comments

All composites in this sequence are 2-pseudoprimes, see A001567, and strong pseudoprimes to base 2, A001262.

The subsequence of these composites begins: 1441091, 3587553971, 4528686251, 23260036451, 47535120323, 61070250323, 90474845819, 143193768587, 162016315907, 173868807611, 180998962187, 238364070323, 285370693931, 298577370323, ...

Perhaps this sequence contains all the terms of the sequence A107007 or A168539.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Subsequence of A176997.

Cf. A003629, A006970, A175865.

Programs

Maple

isA214151 := proc(n)
    if (n mod 6 = 5) and modp(2 &^ ((n-1)/2),n)  = n-1 and modp(3 &^ ((3*n-1)/2),n)  = 3 then
        true;
    else
        false;
    end if;
end proc:
for n from 5 by 6 do
    if isA214151(n) then
        print(n) ;
    end if;
end do: # R. J. Mathar, Jul 20 2012

Mathematica

Select[Range[5,2500,6],PowerMod[3,(3#-1)/2,#]==3&&PowerMod[2,(#-1)/2,#] == #-1&] (* Harvey P. Dale, Mar 14 2022 *)

PARI

for(n=0, 200, b=6*n+5; if(Mod(3, b)^((3*b-1)/2)==3, if(Mod(2, b)^((b-1)/2)==b-1 , print1(b, ", "))));