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A374090 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + x*y + y^2 = k.

%I A374090 #64 Jun 30 2024 22:10:49
%S A374090 0,3,7,147,91,7203,637,352947,1729,24843,31213,847425747,12103,
%T A374090 41523861603,405769,1217307,53599,99698791708803,157339,
%U A374090 4885240793731347,593047,59648043
%N A374090 a(n) is the smallest nonnegative integer k where exactly n ordered pairs of positive integers (x, y) exist such that x^2 + x*y + y^2 = k.
%C A374090 a(n) is the smallest nonnegative k such that A374088(k) = n.
%C A374090 From _Chai Wah Wu_, Jun 28 2024: (Start)
%C A374090 If x <> y and x^2 + x*y + y^2 = a(n), then (x, y) and (y, x) both count as solutions. Therefore if a(n) exists, then a(n) is of the form 3*m^2 if and only if n is odd. This also implies that a(2*n) = A374094(n).
%C A374090 a(25)  = 205724883.
%C A374090 a(27)  = 8968323.
%C A374090 a(33)  = 143214951243.
%C A374090 a(35)  = 10080519267.
%C A374090 a(45)  = 439447827.
%C A374090 a(49)  = 1703607756123.
%C A374090 a(63)  = 21532943523.
%C A374090 a(75)  = 74266682763.
%C A374090 a(81)  = 8618558403.
%C A374090 a(135) = 422309361747.
%C A374090 (End)
%C A374090 From _David A. Corneth_, Jun 29 2024: (Start)
%C A374090 a(19) <= 3*7^18.
%C A374090 a(22) <= 3672178237.
%C A374090 a(24)  = 375193.
%C A374090 a(26)  = 2989441 <= 179936733613.
%C A374090 a(28)  = 29059303.
%C A374090 a(30)  = 7709611.
%C A374090 a(32)  = 1983163.
%C A374090 a(34) <= 432028097404813.
%C A374090 a(36)  = 4877509.
%C A374090 Conjecture: Let q_i be the i-th prime of the form 3*k + 1 and let m = Prod_{j=1, t} b_j, a factorization of m into factors > 1.
%C A374090 Let f(m) = Prod_{j = 1..t} q_i^(b_(t+1-j)-1).
%C A374090 Then for even n we have a(n) = min(f(n), f(n+1))
%C A374090 and for odd n we have a(n) = 3*f(n).
%C A374090 Example for n = 22 we might factor 22 = 11*2. The first two primes of the form 3*k + 1 are 7 and 13. So we would have a(22) = min(7^10*13, 7^22).
%C A374090 a(14) = min(f(14), f(15)) = min(7^6 * 13, 7^4 * 13^2) = 405769. (End)
%F A374090 a(2*n) = A374094(n).
%o A374090 (Python)
%o A374090 from itertools import count
%o A374090 from sympy.abc import x,y
%o A374090 from sympy.solvers.diophantine.diophantine import diop_quadratic
%o A374090 def A374090(n): return next(m for m in (3*k**2 if n&1 else k for k in count(0)) if sum(1 for d in diop_quadratic(x*(x+y)+y**2-m) if d[0]>0 and d[1]>0) == n) # _Chai Wah Wu_, Jun 28 2024
%Y A374090 Cf. A002476, A328151, A374091.
%Y A374090 Cf. A003136, A374088, A374094.
%K A374090 nonn,more
%O A374090 0,2
%A A374090 _Seiichi Manyama_, Jun 28 2024
%E A374090 a(11), a(13) from _Chai Wah Wu_, Jun 28 2024
%E A374090 a(17) from _Bert Dobbelaere_, Jun 28 2024
%E A374090 a(19) from _Bert Dobbelaere_, Jun 30 2024