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A336881
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a(n) is the number of solutions (x, m) of the generalized Ramanujan-Nagell equation x^2 + n = 2^m, x > 0, m > 0, n > 0.
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1
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1, 0, 1, 1, 0, 0, 5, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,7
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COMMENTS
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Equivalently, number of representations of n as n = 2^m - x^2, m > 0, x > 0.
a(7) = 5 corresponds to Ramanujan-Nagell equation (A038198 for x, A060728 for m, Wikipedia link).
If n odd <> 7, R. Apéry proved in 1960 that the equation x^2 + n = 2^m has at most 2 solutions (see link).
If n odd, this equation has 2 solutions iff n = 23 or n = 2^k - 1 for some k >= 4 (link Beukers, theorem 2, p. 395).
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LINKS
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EXAMPLE
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1^2 + 1 = 2^1 hence a(1) = 1.
3^2 + 23 = 2^5 and 45^2 + 23 = 2^11 hence a(23) = 2.
28 = 2^5 - 2^2 = 2^6 - 6^2 = 2^7 - 10^2 = 2^9 - 22^2 = 2^17 - 362^2 hence a(28) = 5.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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