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A247937
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Least integer m > n such that m + n divides F(m) + F(n), where F(k) refers to the Fibonacci number A000045(k).
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16
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5, 22, 9, 8, 8, 18, 10, 16, 21, 14, 35, 24, 17, 34, 21, 32, 20, 30, 31, 28, 87, 26, 47, 36, 28, 46, 63, 32, 80, 42, 151, 40, 75, 38, 38, 60, 113, 39, 51, 56, 109, 49, 307, 52, 63, 50, 50, 72, 101, 70, 57, 68, 97, 66, 58, 64, 93, 62, 191, 84
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OFFSET
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1,1
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COMMENTS
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Conjecture: Let A be any integer not congruent to 3 modulo 6. Define u(0) = 0, u(1) = 1, and u(n+1) = A*u(n) + u(n-1) for n > 0. Then, for any integer n > 0, there are infinitely many positive integers m such that m + n divides u(m) + u(n).
This implies that a(n) exists for any n > 0.
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LINKS
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EXAMPLE
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a(2) = 22 since 22 + 2 = 24 divides F(22) + F(2) = 17711 + 1 = 17712 = 24*738.
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MATHEMATICA
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Do[m=n+1; Label[aa]; If[Mod[Fibonacci[m]+Fibonacci[n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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