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

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

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]

Crossrefs

Intersection of A016813 and A000037.

Cf. A077426, A079896.

Programs

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 *)

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

Python

from operator import sub
from sympy import integer_nthroot
def A077425(n): return n+sub(*integer_nthroot(n,2))<<2|1 # Chai Wah Wu, Oct 01 2024

Extensions

More terms from Max Alekseyev, Mar 03 2010