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

1, 3, 21, 147, 171, 657, 903, 1029, 1197, 2667, 3249, 4599, 6321, 7077, 7203, 8379, 12483, 13203, 18669, 22743, 32193, 38829, 44247, 47961, 49539, 50421, 51471, 58653, 61731, 71631, 87381, 92421, 97641, 113799, 114681, 118341, 130683, 152019, 159201, 197757

Offset: 1

Author

Bernard Schott, Jan 07 2019

Keywords

• nonn
• base

Comments

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.

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

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

Examples

			3 is a term because 3 * 21 = 4^3 - 1.
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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..3200

Crossrefs

Cf. A014945, A024036.

Programs

Maple

filter:= proc(n) local d;
  if 4 &^ n - 1 mod n <> 0 then return false fi;
  for d in select(t -> t > 1 and t^2 <= n, numtheory:-divisors(n)) do
    if 4 &^ d - 1 mod d = 0 and 4 &^ (n/d) - 1 mod (n/d) = 0 then return false fi;
  od;
true
end proc:
select(filter, [$1..200000]); # Robert Israel, Jan 24 2019

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
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

Extensions

More terms (using b-file for A014945) from Jon E. Schoenfield, Jan 11 2019

Terms verified by Jon E. Schoenfield and David A. Corneth, Jan 12 2019