cp's OEIS Frontend

A226693 Squarefree parts of A079896(n), n>= 1.

%I A226693 #26 Jun 02 2025 06:47:50
%S A226693 5,2,3,13,17,5,21,6,7,29,2,33,37,10,41,11,5,3,13,53,14,57,15,61,65,17,
%T A226693 69,2,73,19,77,5,21,85,22,89,23,93,6,97,101,26,105,3,109,7,113,29,13,
%U A226693 30,31,5,2,129,33,133,34,137,35,141,145,37,149,38,17,39,157,10
%N A226693 Squarefree parts of A079896(n), n>= 1.
%C A226693 a(n) is the squarefree part of the discriminant D(n) = A079896(n) of indefinite binary quadratic forms. Certain quadratic irrationals, called omega_p(D(n)), related to the principal indefinite form of discriminant D(n) are integers in the quadratic number field Q(sqrt(a(n))). See A226166 for the definition of these irrationals omega_p(D(n)) using the D. A. Buell reference, p. 31 and p. 26.
%C A226693 For discriminants D == 1 (mod 4) these squarefree parts are given in A226165. For D == 0 (mod 4) the squarefree parts are given in A002734 corresponding to A000037 = D/4.
%D A226693 D. A. Buell, Binary Quadratic Forms, Springer, 1989.
%H A226693 Gheorghe Coserea, <a href="/A226693/b226693.txt">Table of n, a(n) for n = 1..20001</a>
%F A226693 a(n) = squarefree part of D(n) = A079896(n), n >= 1, the numbers 0 and 1 (mod 4), not a square.
%t A226693 SquareFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]); SquareFreePart /@ Select[ Range[160], ! IntegerQ[Sqrt[#]] && Mod[#, 4] < 2 &] (* _Jean-François Alcover_, Jun 25 2013 *)
%o A226693 (PARI)
%o A226693 A079896_list(N) = {
%o A226693   my(n = 1, v = vector(N), top = 0);
%o A226693   while (top < N, if (n%4 < 2 && !issquare(n), v[top++] = n); n++;);
%o A226693   return(v);
%o A226693 };
%o A226693 apply(core, A079896_list(68)) \\ _Gheorghe Coserea_, Nov 10 2016
%Y A226693 Cf. A079896, A226165, A002734.
%K A226693 nonn,easy
%O A226693 1,1
%A A226693 _Wolfdieter Lang_, Jun 15 2013
%E A226693 Offset corrected by _Robin Visser_, Jun 01 2025