A079598 a(n) = 2^(4n+1) - 2^(2n).

1, 28, 496, 8128, 130816, 2096128, 33550336, 536854528, 8589869056, 137438691328, 2199022206976, 35184367894528, 562949936644096, 9007199187632128, 144115187807420416, 2305843008139952128

Offset: 0

Author

Marco Matosic, Jan 28 2003

Keywords

• nonn
• easy

Comments

A sequence rich in perfect numbers, since according to the Euclid-Euler theorem all even perfect numbers are of the form 2^(k-1)*(2^k - 1) and in this sequence k = 2*n + 1.

Let G be a sequence satisfying G(i) = 2*G(i-1) + 8*G(i-2) for arbitrary integers i and without regard to the initial values of G. Then a(n) = (G(i+4*n+2) - G(i)*8^(2*n+1))/(2*G(i+2*n+1)) as long as G(i+2*n+1) != 0. - Klaus Purath, Oct 22 2020

In the binary system, the elements of the sequence consist of a total of 4*n+1 bits starting with 2*n+1 ones followed by 2*n zeros. - Martin Renner, Mar 22 2022

Michael Stifel and Petrus Bungus erroneously claimed that for n > 0 all these numbers were perfect. See links: Bungus, Hassler, and Stifel. - Stefano Spezia, Jul 28 2024

Links

• Petrus Bungus, Numerorum Mysteria, Taupinart, (1618). (In Latin). See p. 468.
• Uwe Hassler, Perfect Numbers, Euleriana: 3(2), pp.176-185, (2023). See pp. 177-178.
• Michael Stifel, Arithmetica integra, Joh. Petreius, (1544). (In Latin). See p. 11.
• László Tóth, Evaluating zeta(s) At Odd Positive Integers Using Automatic Dirichlet Series, arXiv:2508.04151 [math.NT], 2025.
• Index entries for linear recurrences with constant coefficients, signature (20,-64).

Crossrefs

Cf. A000396, A006516.

Programs

Maple

seq(sum((2*k-1)^3,k=1..2^n),n=0..15);
seq(sum(2^k,k=2*n..4*n),n=0..15);

PARI

a(n)=2^(4*n+1)-4^n \\ Charles R Greathouse IV, Nov 29 2011

Formula

a(n+1) = 16*a(n) + 12*2^(2n).

a(n) = Sum_{k=1..2^n} (2*k-1)^3. - Franz Vrabec, Jun 24 2006

a(n) = Sum_{k=2*n..4*n} 2^k. - Martin Renner, Mar 22 2022

G.f.: ( 1+8*x ) / ( (16*x-1)*(4*x-1) ). - R. J. Mathar, Nov 29 2011

E.g.f.: exp(4*x)*(2*exp(12*x) - 1). - Stefano Spezia, Jul 28 2024