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A014945 Numbers k such that k divides 4^k - 1.

%I A014945 #51 Sep 08 2022 08:44:39
%S A014945 1,3,9,21,27,63,81,147,171,189,243,441,513,567,657,729,903,1029,1197,
%T A014945 1323,1539,1701,1971,2187,2667,2709,3087,3249,3591,3969,4599,4617,
%U A014945 5103,5913,6321,6561,7077,7203,8001,8127,8379,9261,9747,10773,11907,12483
%N A014945 Numbers k such that k divides 4^k - 1.
%C A014945 This sequence is closed under multiplication. - _Charles R Greathouse IV_, Nov 03 2016
%C A014945 Conjecture: if k divides 4^k - 1, then (4^k - 1)/k is squarefree. - _Thomas Ordowski_, Dec 24 2018
%C A014945 Following Greathouse's comment, see A323203 for the primitive terms. - _Bernard Schott_, Jan 03 2019
%C A014945 All terms except 1 are divisible by 3.  Proof: suppose n>1 is in the sequence, and let p be its smallest prime factor.  Of course p is odd.  Since 4^n-1 is divisible by p, n is divisible by the multiplicative order of 4 mod p, which is less than p.  But since n has no prime factors < p, that multiplicative order can only be 1, which means p=3. - _Robert Israel_, Jan 24 2019
%H A014945 Amiram Eldar, <a href="/A014945/b014945.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..872 from Muniru A Asiru, terms 873..2000 from Alois P. Heinz)
%F A014945 a(n) = A014741(n+1)/2.
%p A014945 select(n->modp(4^n-1,n)=0,[$1..13000]); # _Muniru A Asiru_, Dec 28 2018
%t A014945 Select[Range[12500],Divisible[4^#-1,#]&]  (* _Harvey P. Dale_, Mar 23 2011 *)
%o A014945 (PARI) is(n)=Mod(4,n)^n==1 \\ _Charles R Greathouse IV_, Nov 03 2016
%o A014945 (GAP) a:=Filtered([1..13000],n->(4^n-1) mod n=0);; Print(a); # _Muniru A Asiru_, Dec 28 2018
%o A014945 (Magma) [n: n in [1..12500] | (4^n-1) mod n eq 0 ]; // _Vincenzo Librandi_, Dec 29 2018
%o A014945 (Python)
%o A014945 for n in range(1,1000):
%o A014945     if (4**n-1) % n ==0:
%o A014945         print(n, end=', ') # _Stefano Spezia_, Jan 05 2019
%Y A014945 Cf. A014741, A323203.
%K A014945 nonn
%O A014945 1,2
%A A014945 _Olivier Gérard_
%E A014945 More terms and better description from _Benoit Cloitre_, Mar 05 2002