A207262 a(n) = 2^(4n - 2) + 1.

5, 65, 1025, 16385, 262145, 4194305, 67108865, 1073741825, 17179869185, 274877906945, 4398046511105, 70368744177665, 1125899906842625, 18014398509481985, 288230376151711745, 4611686018427387905, 73786976294838206465, 1180591620717411303425, 18889465931478580854785, 302231454903657293676545

Offset: 1

Author

Alonso del Arte, Feb 16 2012

Keywords

• nonn
• easy

Comments

With the exception of the first term, all these numbers are composite, and in fact are all multiples of 5. The other factors can be considerably larger than 5, as is the case with say, 2^158 + 1. These numbers can be factored as (2^(2n - 1) + 2^n + 1)(2^(2n - 1) - 2^n + 1). For example, 2^6 + 1 = 65 = (2^3 + 2^2 + 1)(2^3 - 2^2 + 1) = 13 * 5.

This formula was discovered by Leon-Francois-Antoine Aurifeuille in 1873. Wells (2005) remarks that knowledge of this formula would have saved Fortune Landry years of work he spent factoring 2^58 + 1.

Aurifeuille actually rediscovered a very special case of the identity 4x^4+1 = (2x^2-2x+1)(2x^2+2x+1), which Euler communicated to Goldbach in 1742. (The Fuss reference is in my book Seminumerical Algorithms, 3rd ed., p. 392; I had cited Aurifeuille in the 1st and 2nd editions.) - Don Knuth, Feb 09 2013

An Engel expansion of 4 to the base 16 as defined in A181565, with the associated series expansion 4 = 16/5 + 16^2/(5*65) + 16^3/(5*65*1025) + 16^4/(5*65*1025*16385) + .... Cf. A087289 and A199561. - Peter Bala, Oct 29 2013

Conjecture: Let m = 4n - 2. a(n) equals the sum of the m-th powers of the divisors of m divided by the sum of the m-th powers of the odd divisors of m. - Ivan N. Ianakiev, Jan 29 2020

References

• David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005) p. 15

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• FactorDB, Factorizations of 2^(4*n-2)+1
• P. H. Fuss, Correspondance math. et physique, 1 (1843) p. 145.
• Primenumbers Yahoo Group, Aurifeuille and factoring, search results.
• Eric Weisstein's World of Mathematics, Aurifeuillean Factorization.
• Yahoo Groups, Aurifeuille and factoring
• Index entries for linear recurrences with constant coefficients, signature (17,-16).

Crossrefs

Cf. A000051, A052539 (supersets). A016825. A087289, A199561.

Programs

Mathematica

2^(4*Range[20] - 2) + 1
LinearRecurrence[{17, -16}, {5, 65}, 50] (* Vincenzo Librandi, Mar 03 2012 *)

PARI

a(n)=4^(2*n-1)+1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 4^(2n - 1) + 1.

G.f.: 5*x*(1-4*x)/((1-x)*(1-16*x)). - Bruno Berselli, Feb 17 2012

a(1) = 5, a(n) = 16*(a(n-1) - 1) + 1. - Arkadiusz Wesolowski, Feb 17 2012

a(n) = 5*A299960(n-1). - R. J. Mathar, Feb 28 2018

E.g.f.: exp(x) + (exp(16*x) - 5)/4. - Stefano Spezia, Jan 30 2020