A374090 - OEIS

%I #64 Jun 30 2024 22:10:49

%S 0,3,7,147,91,7203,637,352947,1729,24843,31213,847425747,12103,

%T 41523861603,405769,1217307,53599,99698791708803,157339,

%U 4885240793731347,593047,59648043

%N 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 a(n) is the smallest nonnegative k such that A374088(k) = n.

%C From _Chai Wah Wu_, Jun 28 2024: (Start)

%C 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 a(25)  = 205724883.

%C a(27)  = 8968323.

%C a(33)  = 143214951243.

%C a(35)  = 10080519267.

%C a(45)  = 439447827.

%C a(49)  = 1703607756123.

%C a(63)  = 21532943523.

%C a(75)  = 74266682763.

%C a(81)  = 8618558403.

%C a(135) = 422309361747.

%C (End)

%C From _David A. Corneth_, Jun 29 2024: (Start)

%C a(19) <= 3*7^18.

%C a(22) <= 3672178237.

%C a(24)  = 375193.

%C a(26)  = 2989441 <= 179936733613.

%C a(28)  = 29059303.

%C a(30)  = 7709611.

%C a(32)  = 1983163.

%C a(34) <= 432028097404813.

%C a(36)  = 4877509.

%C 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 Let f(m) = Prod_{j = 1..t} q_i^(b_(t+1-j)-1).

%C Then for even n we have a(n) = min(f(n), f(n+1))

%C and for odd n we have a(n) = 3*f(n).

%C 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 a(14) = min(f(14), f(15)) = min(7^6 * 13, 7^4 * 13^2) = 405769. (End)

%F a(2*n) = A374094(n).

%o (Python)

%o from itertools import count

%o from sympy.abc import x,y

%o from sympy.solvers.diophantine.diophantine import diop_quadratic

%o 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 Cf. A002476, A328151, A374091.

%Y Cf. A003136, A374088, A374094.

%K nonn,more

%O 0,2

%A _Seiichi Manyama_, Jun 28 2024

%E a(11), a(13) from _Chai Wah Wu_, Jun 28 2024

%E a(17) from _Bert Dobbelaere_, Jun 28 2024

%E a(19) from _Bert Dobbelaere_, Jun 30 2024