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 A323203 "Primitive" numbers k such that k divides 4^k - 1. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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; 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

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Last modified October 21 11:51 EDT 2021. Contains 348150 sequences. (Running on oeis4.)