

A034017


Numbers that are primitively represented by x^2 + xy + y^2.


15



0, 1, 3, 7, 13, 19, 21, 31, 37, 39, 43, 49, 57, 61, 67, 73, 79, 91, 93, 97, 103, 109, 111, 127, 129, 133, 139, 147, 151, 157, 163, 169, 181, 183, 193, 199, 201, 211, 217, 219, 223, 229, 237, 241, 247, 259, 271, 273, 277, 283, 291, 301, 307, 309, 313, 327, 331
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OFFSET

1,3


COMMENTS

Gives the location of the nonzero terms of A000086.
Starting at a(3), a(n)^2 is the ordered semiperimeter of primitive integer Soddyian triangles (see A210484).  Frank M Jackson, Feb 04 2013
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
The number of structure units in an icosahedral virus is 20*a(n), see Stannard link.  Charles R Greathouse IV, Nov 03 2015
From Wolfdieter Lang, Apr 09 2021: (Start)
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).
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, ..., k1}. 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, 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.
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.
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.
The table with the A341422(n) solutions j of the congruence given above are given in A343232. (End)
Apparently, also the integers k that can be expressed as a quotient of two terms from A002061.  Martin Becker, Aug 14 2022


REFERENCES

B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, see p. 184, AMS, Providence, RI, 1995.
D. A. Buell, Binary Quadratic Forms, Springer, 1989, pp. 15, 19.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Linda Stannard, Principles of Virus Architecture (1995).


FORMULA

n >= 2: 3^{0 or 1} X product of primes of form 3a+1 (A002476) to any nonnegative power.
The sequence {a(n)}_{n>=2} gives the increasingly sorted positive numbers k such that the set M(k) := {j = 0, 1, 2, ... , k1  j^2 + j + 1 == 0 (mod k)}, has cardinality >= 1.  Wolfdieter Lang, Apr 09 2021


MAPLE

N:= 1000: # to get all terms <= N
P:= select(isprime, [seq(6*n+1, n=1..floor((N1)/6))]):
A:= {1, 3}:
for p in P do
A:= {seq(seq(a*p^k, k=0..floor(log[p](N/a))), a=A)}:
od:
sort(convert(A, list)); # Robert Israel, Nov 04 2015


MATHEMATICA

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 *)


PROG

(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
(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
(Haskell)
a034017 n = a034017_list !! (n1)
a034017_list = 0 : filter ((> 0) . a000086) [1..]
 Reinhard Zumkeller, Jun 23 2013


CROSSREFS

Cf. A000217, A002061, A002476, A003136, A007645 (primes), A045611, A045897, A226946 (complement), A045897 (subsequence), A341422, A343232.
Sequence in context: A353357 A352140 A258117 * A034021 A216516 A310260
Adjacent sequences: A034014 A034015 A034016 * A034018 A034019 A034020


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Extended by Ray Chandler, Jan 29 2009


STATUS

approved



