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A122780
Nonprimes k such that 3^k == 3 (mod k).
16
1, 6, 66, 91, 121, 286, 561, 671, 703, 726, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7107, 7381, 8205, 8401, 8646, 8911, 10585, 11011, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345
OFFSET
1,2
COMMENTS
Theorem: If q!=3 and both numbers q and (2q-1) are primes then k=q*(2q-1) is in the sequence. 6, 91, 703, 1891, 2701, 12403, 18721, 38503, 49141, ... is the related subsequence.
The terms > 1 and coprime to 3 of this sequence are the base-3 pseudoprimes, A005935. - M. F. Hasler, Jul 19 2012 [Corrected by Jianing Song, Feb 06 2019]
LINKS
EXAMPLE
66 is composite and 3^66 = 66*468229611858069884271524875811 + 3 so 66 is in the sequence.
MAPLE
isA122780 := proc(n)
if isprime(n) then
false;
else
modp( 3 &^ n, n) = modp(3, n) ;
end if;
end proc:
for n from 1 do
if isA122780(n) then
print(n) ;
end if;
end do: # R. J. Mathar, Jul 15 2012
MATHEMATICA
Select[Range[30000], ! PrimeQ[ # ] && Mod[3^#, # ] == Mod[3, # ] &]
Join[{1}, Select[Range[20000], !PrimeQ[#]&&PowerMod[3, #, #]==3&]] (* Harvey P. Dale, Apr 30 2023 *)
PROG
(PARI) is_A122780(n)={n>0 & Mod(3, n)^n==3 & !ispseudoprime(n)} \\ M. F. Hasler, Jul 19 2012
KEYWORD
easy,nonn
AUTHOR
Farideh Firoozbakht, Sep 11 2006
STATUS
approved