|
%I #72 Feb 26 2024 10:12:00
%S 0,1,3,7,13,19,21,31,37,39,43,49,57,61,67,73,79,91,93,97,103,109,111,
%T 127,129,133,139,147,151,157,163,169,181,183,193,199,201,211,217,219,
%U 223,229,237,241,247,259,271,273,277,283,291,301,307,309,313,327,331
%N Numbers that are primitively represented by x^2 + xy + y^2.
%C Gives the location of the nonzero terms of A000086.
%C Starting at a(3), a(n)^2 is the ordered semiperimeter of primitive integer Soddyian triangles (see A210484). - _Frank M Jackson_, Feb 04 2013
%C A000086(a(n)) > 0; a(n) = A004611(k) or a(n) = 3*A004611(k) for n > 3 and an appropriate k. - _Reinhard Zumkeller_, Jun 23 2013
%C The number of structure units in an icosahedral virus is 20*a(n), see Stannard link. - _Charles R Greathouse IV_, Nov 03 2015
%C From _Wolfdieter Lang_, Apr 09 2021: (Start)
%C The positive definite binary quadratic form F = [1, 1, 1], that is x^2 + x*y + y^2, has discriminant Disc = -3, and class number 1 (see Buell, Examples, p. 19, first line: Delta = -3, h = 1). This reduced form is equivalent to the form [1,-1, 1], but to no other reduced one (see Buell, Theorem 2.4, p. 15).
%C This form F represents a positive integer k (= a(n)) properly if and only if A002061(j+1) = 2*T(j) + 1 = j^2 + j + 1 == 0 (mod k), for j from {0, 1, ..., k-1}. This congruence determines the representative parallel primitive forms (rpapfs) of discriminant Disc = -3 and representation of a positive integer number k, given by [k, 2*j+1, c(j)], and c(j) is determined from Disc =-3 as c(j) = ((2*j+1)^2 + 3)/(4*k) = (j^2 + j + 1)/k. Each rpapf has a first reduced form, the so-called right neighbor form, namely [1, 1, 1] for k = 1 = a(1) (the already reduced parallel form from j = 0), and [1, -1, 1] for k = a(n), with n >= 2.
%C Only odd numbers k are eligible for representation, because 2*T(j) + 1, with the triangular numbers T = A000217, is odd. The odd k with at least one solution of the congruence are then the members of the present sequence.
%C The solutions of the reduced forms F = [1, 1, 1] and F' = [1, -1, 1] representing k are related by type I equivalence because of the first two entries ([a, a, c] == [a, -a, c]), and also by type II equivalence because [a, b, a] == [a, -b, a], for positive b. These transformation matrices are R_I = Matrix([1, -1],[0, 1]) and R_{II} = Matrix([0, -1], [1, 0]), respectively, to obtain the forms with negative second entry from the ones with positive second entry. The corresponding solutions (x, y)^t (t for transposed) are related by the inverse of these matrices.
%C The table with the A341422(n) solutions j of the congruence given above are given in A343232. (End)
%C Apparently, also the integers k that can be expressed as a quotient of two terms from A002061. - _Martin Becker_, Aug 14 2022
%C For some x, y let a(n) = r, x*(x+y) = s, y*(x+y) = t, x*y = u then (r,s,t,u) is a Pythagorean quadruple such that r^2 = s^2 + t^2 + u^2. - _Frank M Jackson_, Feb 26 2024
%D B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, see p. 184, AMS, Providence, RI, 1995.
%D D. A. Buell, Binary Quadratic Forms, Springer, 1989, pp. 15, 19.
%H Reinhard Zumkeller, <a href="/A034017/b034017.txt">Table of n, a(n) for n = 1..10000</a>
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H Linda Stannard, <a href="http://www.virology.uct.ac.za/vir/teaching/linda-stannard/principles-of-virus-architecture">Principles of Virus Architecture</a> (1995).
%F n >= 2: 3^{0 or 1} X product of primes of form 3a+1 (A002476) to any nonnegative power.
%F The sequence {a(n)}_{n>=2} gives the increasingly sorted positive numbers k such that the set M(k) := {j = 0, 1, 2, ..., k-1 | j^2 + j + 1 == 0 (mod k)}, has cardinality >= 1. - _Wolfdieter Lang_, Apr 09 2021
%p N:= 1000: # to get all terms <= N
%p P:= select(isprime, [seq(6*n+1, n=1..floor((N-1)/6))]):
%p A:= {1,3}:
%p for p in P do
%p A:= {seq(seq(a*p^k, k=0..floor(log[p](N/a))),a=A)}:
%p od:
%p sort(convert(A,list)); # _Robert Israel_, Nov 04 2015
%t lst = {0}; maxLen = 331; Do[If[Reduce[m^2 + m*n + n^2 == k && m >= n >= 0 && GCD[m, n] == 1, {m, n}, Integers] === False, , AppendTo[lst, k]], {k, maxLen}]; lst (* _Frank M Jackson_, Jan 10 2013 *) (* simplified by _T. D. Noe_, Feb 05 2013 *)
%o (PARI) is(n)=my(f=factor(n));for(i=1,#f[,1],if(f[i,1]%3!=1 && (f[i,1]!=3 || f[i,2]>1), return(n==0))); 1 \\ _Charles R Greathouse IV_, Jan 10 2013
%o (PARI) list(lim)=if(lim<7, return(select(n->n<=lim, [0,1,3]))); my(v=List([0,1,3])); for(x=1,sqrtint(lim\=1), my(y,t); while(y++<x && (t=x^2+x*y+y^2)<=lim, gcd(x,y)==1 && listput(v,t))); Set(v) \\ _Charles R Greathouse IV_, Jan 20 2022
%o (Haskell)
%o a034017 n = a034017_list !! (n-1)
%o a034017_list = 0 : filter ((> 0) . a000086) [1..]
%o -- _Reinhard Zumkeller_, Jun 23 2013
%Y Cf. A000217, A002061, A002476, A003136, A007645 (primes), A045611, A045897, A226946 (complement), A045897 (subsequence), A341422, A343232.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_
%E Extended by _Ray Chandler_, Jan 29 2009
|
