A300419 Smallest nonnegative number k such that k can be written in exactly n ways as x^2 + xy + y^2 where x and y are positive integers, with x >= y.

0, 3, 91, 637, 1729, 24843, 12103, 405769, 53599, 157339, 593047, 59648043, 375193, 2989441, 8968323, 7709611, 1983163, 3360173089, 4877509, 2339177536969, 18384457, 377770939, 146482609, 439447827, 13882141, 1302924259

Offset: 0

Author

Altug Alkan, M. F. Hasler and Robert G. Wilson, Mar 05 2018

Keywords

• nonn
• more

Comments

Except a(0) and a(1), all terms are in A118886.

First positive square term of this sequence is a(7) = 405769 = a(3)^2.

a(3), a(7) = a(3)^2 and a(13) = a(4)^2 are also the sum of two nonzero squares in exactly one way.

a(18) = 4877509, a(20) = 18384457, a(22) = 146482609, a(24) = 13882141, a(27) = 92672671, a(30) = 238997941, a(32) = 85276009, a(36) = 180467833. - Robert G. Wilson v, Mar 06 2018

Examples

			a(2) = 91 because 91 = 1^2 + 1*9 + 9^2 = 5^2 + 5*6 + 6^2 and 91 is the least number with this property.
		

Links

• Robert G. Wilson v, Solutions of a(n) for n <= 16

Crossrefs

Cf. A024614, A118886, A198799, A327796, A327797.

Programs

Mathematica

nmx = 4750; t = Split@ Sort@ Flatten@ Table[x^2 + x*y + y^2, {x, nmx}, {y, x, nmx}]; lmt = 1 + Length@ t; f[n_] := Block[{k = 1}, While[Length@ t[[k]] != n && k < lmt, k++]; t[[k]][[1]]]; Array[f, 16] (* Robert G. Wilson v, Mar 06 2018 *)

PARI

N(n,d)=sum(x=1,sqrt(n\3),sum(y=max(x,sqrtint(n-x^2)\2),sqrtint(n-2*x^2),x^2+x*y+y^2==n&&!(d&&printf("%d",[x,y])))) \\ Set 2nd arg = 1 to display all decompositions.
a(n)=for(k=0,oo,N(k)==n&&return(k))

Formula

If A198799(n) is not a square and there is no square s < A198799(n) such that A088534(s) = n + 1, then a(n) = A198799(n), for all n > 0.

If A198799(n+1) is a square, then a(n) <= A198799(n+1).

Extensions

a(17)-a(18) from Giovanni Resta, Mar 16 2018

a(19)-a(25) from Bert Dobbelaere, Feb 18 2023