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a(n) = 3*(6858365065530*(2^45 - 1)*n + 153479820268467961)^2.
3

%I #33 Oct 31 2023 11:21:17

%S 70668165688923686196507258250492563,

%T 174687593550891106640307045856561008882907291372256643,

%U 698750373759134872171732581703201135992894186495330123,1572188340624731296664944773228844067526467943619713003

%N a(n) = 3*(6858365065530*(2^45 - 1)*n + 153479820268467961)^2.

%C The "1/k" heuristic predicts that primes of the form k*2^m + 1 with k odd and m > 0 have almost a 1/k chance of being Fermat divisors (Dubner and Keller). This sequence yields a correction to the "1/k" heuristic, because it generates special values of k.

%C If:

%C 1) k is of the form 3*a^2, where a is an odd positive integer not divisible by 3,

%C 2) k is not a SierpiƄski number,

%C 3) for all odd positive integers m the numbers k*2^m + 1 are composite,

%C then the probability that a Fermat number is divisible by a prime of the form k*2^m + 1 equal to 0.

%C Every term meets the first and third condition. For any n, at least one of the primes from A361898 (except 3) divides every integer in the sequence a(n)*2^m + 1 with m odd.

%C What is the smallest odd integer k such that every prime of the form k*2^m + 1 (m > 0) does not divide any Fermat number?

%D H. Suyama, A note on the factors of Fermat numbers II, Abstracts of Papers Presented to the Amer. Math. Soc., Vol. 5 (1984), p. 132.

%H Arkadiusz Wesolowski, <a href="/A361899/b361899.txt">Table of n, a(n) for n = 0..10000</a>

%H Harvey Dubner and Wilfrid Keller, <a href="http://dx.doi.org/10.1090/S0025-5718-1995-1270618-1">Factors of Generalized Fermat Numbers</a>, Math. Comp. 64 (1995), no. 209, 397-405.

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2005337">Properties of the sequence 3.2^n + 1</a>, Math. Comp., 30 (1976), 657-663.

%H S. W. Golomb, <a href="/A004119/a004119.pdf">Properties of the sequence 3.2^n + 1</a>, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]

%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_1145.htm">Puzzle 1145. Divisors of Fermat numbers</a>, The Prime Puzzles and Problems Connection.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (70668165688923686196507258250492563 + 174687593550891106428302548789789950293385516620778954*x + 174687593106461552462815941200289167933694087130037883*x^2)/(1 - x)^3.

%F a(n) = 3*(2*(Product_{i=1..13} A361898(i))*n + 153479820268467961)^2.

%F a(n) = 3*((29062/1192737)*(2^48 - 1)*(2^45 - 1)*n + 153479820268467961)^2.

%t Table[3 (6858365065530 (2^45 - 1) n + 153479820268467961)^2, {n, 0, 3}]

%o (Magma) [3*(6858365065530*(2^45-1)*n+153479820268467961)^2: n in [0..3]];

%o (PARI) a(n)=3*(6858365065530*(2^45-1)*n+153479820268467961)^2

%Y Cf. A000215, A229852, A351332, A361898, A361900.

%K nonn,easy

%O 0,1

%A _Arkadiusz Wesolowski_, Mar 28 2023