A024614 Numbers of the form x^2 + xy + y^2, where x and y are positive integers.

3, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199, 201, 208, 211, 217, 219, 223, 228

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Equivalently, sequence A024612 with duplicates removed; i.e., numbers of the form i^2 - i*j + j^2, where 1 <= i < j.

A subsequence of A135412, which consists of multiples (by squarefree factors) of the numbers listed here. It appears that this lists numbers > 1 which have in their factorization: (a) no even power of 3 unless there is a factor == 1 (mod 6); (b) no odd power of 2 or of a prime == 5 (mod 6) and no even power unless there is a factor 3 or == 1 (mod 6). - M. F. Hasler, Aug 17 2016

If we regroup the entries in a triangle with row lengths A004526

3,

7,

12, 13,

19, 21,

27, 28, 31,

37, 39, 43,

... it seems that the j-th row of the triangle contains the numbers i^2+j^2-i*j in row j>=2 and column i = floor((j+1)/2) .. j-1. - R. J. Mathar, Aug 21 2016

Proof of the above characterization: the sequence is the union of 3*(the squares A000290) and A024606 (numbers x^2+xy+y^2 with x > y > 0). For the latter it is known that these are the numbers with a factor p==1 (mod 6) and any prime factor == 2 (mod 3) occurring to an even power. The former (3*n^2) are the same as (odd power of 3)*(even power of any other prime factor). The union of the two cases yields the earlier characterization. - M. F. Hasler, Mar 04 2018

Least term that can be written in exactly n ways is A300419(n). - Altug Alkan, Mar 04 2018

For the general theory see the Fouvry et al. reference and A296095. Bounds used in the Julia program are based on the theorems in this paper. - Peter Luschny, Mar 10 2018

Examples

			3 = 1^2 + 1^2 + 1*1, 7 = 2^2 + 1^2 + 2*1, ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Peter Luschny)
• Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Crossrefs

Cf. A003136, A007645 (prime terms), A024612, A135412, A296095, A300419.

Programs

Julia

function isA024614(n)
    n % 3 >= 2 && return false
    n == 3 && return true
    M = Int(round(2*sqrt(n/3)))
    for y in 2:M, x in 1:y
        n == x^2 + y^2 + x*y && return true
    end
    return false
end
A024614list(upto) = [n for n in 1:upto if isA024614(n)]
println(A024614list(228)) # Peter Luschny, Mar 02 2018 updated Mar 17 2018

Maple

isA024614 := proc(n)
    local i,j,disc;
    # n=i^2+j^2-i*j = (j-i)^2+i*j, 1<=i=1 and i*j>=j and i^2+j^2-i*j >= 1+j max search radius
    for j from 2 to n-1 do
        # i=(j +- sqrt(4n-3j^2))/2
        disc := 4*n-3*j^2 ;
        if disc >= 0 then
            if issqr(disc) then
                i := (j+sqrt(disc))/2 ;
                if type(i,'integer') and i >= 1 and i= 1 and iA024614(t) then
        printf("%d %d\n",n,t) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 21 2016
# second Maple program:
a:= proc(n) option remember; local k, x;
      for k from a(n-1)+1 do for x while x^2 x^2+(x+y)*y=k)((isqrt(4*k-3*x^2)-x)/2) then return k fi
      od od
    end: a(0):=0:
seq(a(n), n=1..200);  # Alois P. Heinz, Mar 02 2018

Mathematica

max = 228; T0 = {}; xm = Ceiling[Sqrt[max]]; While[T = T0; T0 = Table[x^2 + x y + y^2, {x, 1, xm}, {y, x, xm}] // Flatten // Union // Select[#, # <= max&]&; T != T0, xm = 2 xm]; T (* Jean-François Alcover, Mar 23 2018 *)

PARI

is(n)={n>2&&!for(i=1, #n=Set(Col(factor(n)%6))/*consider prime factors mod 6*/, n[i][1]>1||next/*skip factors = 1 mod 6*/; /* odd power: ok only if p=3 */n[i][2]%2&&(n[i][1]!=3 || next) && return; /*even power: ok if there's a p==1, listed first*/ n[1][1]==1 || /*also ok if it's not a 3 and if there's a 3 elsewhere */ (n[i][1]==2 && i<#n && n[i+1][1]==3) || (n[i][1]>3 && for(j=1,i-1,n[j][1]==3 && next(2))||return))} \\ M. F. Hasler, Aug 17 2016, documented & bug fixed (following an observation by Altug Alkan) Mar 04 2018

PARI

is(n)={(n=factor(n))||return/*n=1*/; /*exponents % 2, primes % 3*/ n[,2]%=2; n[,1]%=3; (n=Set(Col(n))) /*odd power of a prime == 2? will be last*/ [#n][2] && n[#n][1]==2 && return; /*factor == 1? will be 1st or after 3*/ n[1+(!n[1][1] && #n>1)][1]==1 || /*thrice a square?*/ (!n[1][1]&&n[1][2]&&!for(i=2,#n,n[i][2]&&return))} \\ Alternate code, 5% slower, maybe a bit less obscure. - M. F. Hasler, Mar 04 2018

PARI

N=228; v=vector(N);
for(x=1,N, x2=x*x; if(x2>N,break); for(y=x,N, t=x2+y*(x+y); if(t>N,break); v[t]=1;));
for(n=1,N,if(v[n],print1(n,", "))); \\ Joerg Arndt, Mar 10 2018

PARI

list(lim)=my(v=List(), x2); lim\=1; for(x=1, sqrtint(4*lim\3), x2=x^2; for(y=1, min((sqrt(4*lim-3*x2)-x)/2, x), listput(v, y*(x+y)+x2))); Set(v) \\ Charles R Greathouse IV, Mar 23 2018

Extensions

Edited by M. F. Hasler, Aug 17 2016

b-file for values a(1)..a(10^4) double-checked with PARI code by M. F. Hasler, Mar 04 2018