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
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