cp's OEIS Frontend

A323203 "Primitive" numbers k such that k divides 4^k - 1.

%I A323203 #51 Sep 11 2019 08:45:17
%S A323203 1,3,21,147,171,657,903,1029,1197,2667,3249,4599,6321,7077,7203,8379,
%T A323203 12483,13203,18669,22743,32193,38829,44247,47961,49539,50421,51471,
%U A323203 58653,61731,71631,87381,92421,97641,113799,114681,118341,130683,152019,159201,197757
%N A323203 "Primitive" numbers k such that k divides 4^k - 1.
%C A323203 In the comments of A014945, Charles R. Greathouse writes "this sequence is closed under multiplication". So, here, the terms are only the "primitive" integers which satisfy the definition and are not the product of two or more previous numbers of the sequence. This sequence is a subsequence of A014945.
%C A323203 Also numbers k in A014945 such that no divisors d > 1 of k exist where d and k/d are in A014945. - _David A. Corneth_, Jan 11 2019
%C A323203 Following an observation of David A. Corneth, yes, a(n) is divisible by 3 for n > 1, there is a proof by _Robert Israel_ in A014945. - _Bernard Schott_, Jan 25 2019
%H A323203 Amiram Eldar, <a href="/A323203/b323203.txt">Table of n, a(n) for n = 1..3200</a>
%e A323203 3 is a term because 3 * 21 = 4^3 - 1.
%e A323203 63 divides 4^63 - 1, but 63 is not a term because 63 = 3 * 21 with 3 which divides 4^3 - 1, and 21 which divides 4^21 - 1.
%p A323203 filter:= proc(n) local d;
%p A323203   if 4 &^ n - 1 mod n <> 0 then return false fi;
%p A323203   for d in select(t -> t > 1 and t^2 <= n, numtheory:-divisors(n)) do
%p A323203     if 4 &^ d - 1 mod d = 0 and 4 &^ (n/d) - 1 mod (n/d) = 0 then return false fi;
%p A323203   od;
%p A323203 true
%p A323203 end proc:
%p A323203 select(filter, [$1..200000]); # _Robert Israel_, Jan 24 2019
%o A323203 (PARI) is(n) = my(d=divisors(n)); if(Mod(4,n)^n != 1, return(0)); for(i = 2, (#d - 1) >> 1 + 1, if(Mod(4,d[i]) ^ d[i] == 1 && Mod(4, n/d[i]) ^ (n/d[i])==1, return(0))); 1
%o A323203 first(n) = n = max(n, 2); my(res = vector(n), t=1); res[1] = 1;forstep(i = 3, oo, 3, if(is(i), t++; res[t] = i; if(t==n, return(res)))) \\ _David A. Corneth_, Jan 11 2019
%Y A323203 Cf. A014945, A024036.
%K A323203 nonn,base
%O A323203 1,2
%A A323203 _Bernard Schott_, Jan 07 2019
%E A323203 More terms (using b-file for A014945) from _Jon E. Schoenfield_, Jan 11 2019
%E A323203 Terms verified by _Jon E. Schoenfield_ and _David A. Corneth_, Jan 12 2019