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

1,2

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

Links

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

Example

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.

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

Prog

(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

Crossrefs

Cf. A014945, A024036.

Sequence in context: A141492 A243397 A173350 * A169634 A088088 A206638

Adjacent sequences: A323200 A323201 A323202 * A323204 A323205 A323206

Keyword

nonn,base

Author

Bernard Schott, Jan 07 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

Status

approved