A374090 revision #61
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A374090
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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.
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3
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0, 3, 7, 147, 91, 7203, 637, 352947, 1729, 24843, 31213, 847425747, 12103, 41523861603, 405769, 1217307, 53599, 99698791708803, 157339, 4885240793731347, 593047, 59648043
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OFFSET
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0,2
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COMMENTS
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a(n) is the smallest nonnegative k such that A374088(k) = n.From Chai Wah Wu, Jun 28 2024: (Start)
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).
a(25) = 205724883.
a(27) = 8968323.
a(33) = 143214951243.
a(35) = 10080519267.
a(45) = 439447827.
a(49) = 1703607756123.
a(63) = 21532943523.
a(75) = 74266682763.
a(81) = 8618558403.
a(135) = 422309361747.
(End)
a(19) <= 3*7^18.
a(22) <= 3672178237.
a(24) = 375193.
a(26) = 2989441 <= 179936733613.
a(28) = 29059303.
a(30) = 7709611.
a(32) = 1983163.
a(34) <= 432028097404813.
a(36) = 4877509.
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.
Let f(m) = Prod_{j = 1..t} q_i^(b_(t+1-j)-1).
Then for even n we have a(n) = min(f(n), f(n+1))
and for odd n we have a(n) = 3*f(n).
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).
a(14) = min(f(14), f(15)) = min(7^6 * 13, 7^4 * 13^2) = 405769. (End)
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LINKS
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FORMULA
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PROG
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(Python)
from itertools import count
from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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proposed
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