{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

User: {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

{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's wiki page.

{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 has authored 1 sequences.

A034017 Numbers that are primitively represented by x^2 + xy + y^2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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, ..., 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.
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
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

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.

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).

Cf. A000217, A002061, A002476, A003136, A007645 (primes), A045611, A045897, A226946 (complement), A045897 (subsequence), A341422, A343232.

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

N:= 1000: # to get all terms <= N
P:= select(isprime, [seq(6*n+1, n=1..floor((N-1)/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

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 *)

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

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++Charles R Greathouse IV, Jan 20 2022

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, ..., k-1 | j^2 + j + 1 == 0 (mod k)}, has cardinality >= 1. - _Wolfdieter Lang, Apr 09 2021

Extended by Ray Chandler, Jan 29 2009

cp's OEIS Frontend

User: {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

A034017 Numbers that are primitively represented by x^2 + xy + y^2.

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