A361899 a(n) = 3*(6858365065530*(2^45 - 1)*n + 153479820268467961)^2.

70668165688923686196507258250492563, 174687593550891106640307045856561008882907291372256643, 698750373759134872171732581703201135992894186495330123, 1572188340624731296664944773228844067526467943619713003

Offset: 0

Author

Arkadiusz Wesolowski, Mar 28 2023

Keywords

• nonn
• easy

Comments

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.

If:

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

2) k is not a Sierpiński number,

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

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

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.

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?

References

• 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.

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 0..10000
• Harvey Dubner and Wilfrid Keller, Factors of Generalized Fermat Numbers, Math. Comp. 64 (1995), no. 209, 397-405.
• S. W. Golomb, Properties of the sequence 3.2^n + 1, Math. Comp., 30 (1976), 657-663.
• S. W. Golomb, Properties of the sequence 3.2^n + 1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
• Carlos Rivera, Puzzle 1145. Divisors of Fermat numbers, The Prime Puzzles and Problems Connection.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

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

Programs

Magma

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

Mathematica

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

PARI

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

Formula

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

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

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