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

%I A077425 #55 Oct 01 2024 03:39:01
%S A077425 5,13,17,21,29,33,37,41,45,53,57,61,65,69,73,77,85,89,93,97,101,105,
%T A077425 109,113,117,125,129,133,137,141,145,149,153,157,161,165,173,177,181,
%U A077425 185,189,193,197,201,205,209,213,217,221,229,233,237,241,245,249,253,257
%N A077425 a(n) == 1 (mod 4) (see A016813), but not a square (i.e., not in A000290).
%C A077425 The Pell equation x^2 - a(n)*y^2 = +4 has infinitely many (integer) solutions (see A077428 and A078355).
%C A077425 These are the odd numbers in A079896. The even ones are 4*A000037. - _Wolfdieter Lang_, Sep 15 2015
%C A077425 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
%C A077425 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
%H A077425 Vincenzo Librandi, <a href="/A077425/b077425.txt">Table of n, a(n) for n = 1..1000</a>
%H A077425 S. R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a> [Cached copy, with permission of the author]
%H A077425 A. M. Legendre, <a href="https://books.google.fr/books?id=EBtPMqDlPxcC&amp;pg=PA476">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</a>, Essai sur la Théorie des Nombres An VI, Table II. [_Paul Curtz_, Apr 11 2019]
%p A077425 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:
%p A077425 seq(A077425(n),n=1..31) ; # _R. J. Mathar_, Apr 25 2006
%t A077425 Select[Range[5,300,4],!IntegerQ[Sqrt[#]]&] (* _Harvey P. Dale_, Dec 05 2012 *)
%o A077425 (PARI) [n | n <- vector(100,n,4*n+1), !issquare(n)] \\ _Charles R Greathouse IV_, Mar 11 2014
%o A077425 (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
%o A077425 (Python)
%o A077425 from operator import sub
%o A077425 from sympy import integer_nthroot
%o A077425 def A077425(n): return n+sub(*integer_nthroot(n,2))<<2|1 # _Chai Wah Wu_, Oct 01 2024
%Y A077425 Intersection of A016813 and A000037.
%Y A077425 Cf. A077426, A079896.
%K A077425 nonn,easy
%O A077425 1,1
%A A077425 _Wolfdieter Lang_, Nov 29 2002
%E A077425 More terms from _Max Alekseyev_, Mar 03 2010