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A110177
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Number of solutions 0<k<n to the equation sigma(n) = sigma(k) + sigma(n-k), where sigma is the sum of divisors function.
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3
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0, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 2, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 2, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 2, 0, 2, 0, 2, 2, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 4
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OFFSET
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1,3
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COMMENTS
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The number of solutions is always even because k=n/2 cannot be a solution for even n.
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LINKS
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MATHEMATICA
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a[n_] := Select[Range[n-1], DivisorSigma[1, n]==DivisorSigma[1, n-# ]+DivisorSigma[1, # ]&]; Table[Length[a[n]], {n, 150}]
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PROG
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(PARI) A110177(n) = { my(x=sigma(n)); sum(k=1, n-1, (x == (sigma(k)+sigma(n-k)))); }; \\ Antti Karttunen, Feb 20 2023
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CROSSREFS
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Cf. A110176 (least k such that sigma(n)=sigma(k)+sigma(n-k)).
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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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