A004770 Numbers of the form 8k+5; or, numbers whose binary expansion ends in 101.

5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93, 101, 109, 117, 125, 133, 141, 149, 157, 165, 173, 181, 189, 197, 205, 213, 221, 229, 237, 245, 253, 261, 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 373, 381, 389, 397, 405, 413, 421, 429, 437, 445

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Only numbers of the form 8k+5 may be written as a sum of 5 odd squares. Examples: 5 = 1+1+1+1+1, 13 = 9+1+1+1+1, 21 = 9+9+1+1+1, 29 = 25+1+1+1+1= 9+9+9+1+1, 37 = 9+9+9+9+1 = 25+9+1+1+1, 45 = 25+9+9+1+1=9+9+9+9+9, 53 = 49+1+1+1+1 = 25+25+1+1+1 = 25+9+9+9+1, ... - Philippe Deléham, Sep 03 2005

Positive solutions to the equation x == 5 (mod 8). - K.V.Iyer, Apr 27 2009

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 248.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• Tanya Khovanova, Recursive Sequences.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A016813, A017101, A017113, A110534, A113770.

Cf. A004776 (complement), A007521 (primes).

Programs

Haskell

a004770 = (subtract 3) . (* 8)
a004770_list = [5, 13 ..]  -- Reinhard Zumkeller, Aug 17 2012

Magma

[8*n-3: n in [1..60]]; // Vincenzo Librandi, May 28 2011

Mathematica

Range[5, 1000, 8] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)
LinearRecurrence[{2,-1},{5,13},60] (* or *) NestList[#+8&,5,60] (* Harvey P. Dale, Jun 28 2021 *)

PARI

a(n)=8*n-3 \\ Charles R Greathouse IV, Sep 24 2015

SageMath

[8*n-3 for n in range(1,57)] # Stefano Spezia, Jul 23 2025

Formula

From R. J. Mathar, Mar 14 2011: (Start)

a(n) = 8*n - 3.

G.f.: x*(5+3*x)/(x-1)^2. (End)

a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, May 28 2011

From Elmo R. Oliveira, Apr 03 2025: (Start)

E.g.f.: exp(x)*(8*x - 3) + 3.

a(n) = A113770(n)/2 = A016813(2*n-1). (End)