A077425 a(n) == 1 (mod 4) (see A016813), but not a square (i.e., not in A000290).

5, 13, 17, 21, 29, 33, 37, 41, 45, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 117, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 245, 249, 253, 257

Offset

1,1

Comments

The Pell equation x^2 - a(n)*y^2 = +4 has infinitely many (integer) solutions (see A077428 and A078355).

These are the odd numbers in A079896. The even ones are 4*A000037. - Wolfdieter Lang, Sep 15 2015

First differences: 8, 4, 4, 8, 4, 4, 4, 4, 8, 4, 4, 4, 4, 4, 4, 8, ... , only 4's and 8's?. - Paul Curtz, Apr 11 2019

Yes. There are only 4's and 8's. Proof: Only multiples of 4 may appear. The 4's correspond to successive composite in A016813, whereas an 8 corresponds to a square. A greater multiple of 4 would imply to have at least 2 consecutive squares in A016813, which is not possible since 2 consecutive squares cannot have a difference of 4. That sequence of 4's and 8's can be obtained with A010052 (without the 1st term) where the 0's are replaced with 4's and 1's replaced with 8's. - Michel Marcus, Apr 16 2019

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

S. R. Finch, Class number theory [Cached copy, with permission of the author]

A. M. Legendre, Expression les plus simples des formules Ly^2+Myz+Nz^2 où M est impair pour toutes les valeurs de B = M^2-4LN depuis B=5 jusqu'à B=305, Essai sur la Théorie des Nombres An VI, Table II. [Paul Curtz, Apr 11 2019]

Maple

A077425 := proc(n::integer) local resul, i ; resul := 5 ; i := 1 ; while i < n do resul := resul+4 ; while issqr(resul) do resul := resul+4 ; od ; i:= i+1 ; od ; RETURN(resul) ; end proc:
seq(A077425(n), n=1..31) ; # R. J. Mathar, Apr 25 2006

Mathematica

Select[Range[5, 300, 4], !IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Dec 05 2012 *)

Prog

(PARI) [n | n <- vector(100, n, 4*n+1), !issquare(n)] \\ Charles R Greathouse IV, Mar 11 2014
(PARI) list(lim)=my(v=List()); for(s=2, sqrtint((lim\=1)+1), forstep(n=s^2 + if(s%2, 4, 1), min((s+1)^2-1, lim), 4, listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Nov 04 2021

Crossrefs

Intersection of A016813 and A000037.

Cf. A077426, A079896.

Sequence in context: A174361 A226165 A166409 * A039955 A213340 A014539

Adjacent sequences: A077422 A077423 A077424 * A077426 A077427 A077428

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 29 2002

Extensions

More terms from Max Alekseyev, Mar 03 2010

Status

approved