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A337101
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Number of partitions of n into two positive parts (s,t), s <= t, such that the harmonic mean of s and t is an integer.
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5
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0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 2, 2, 1, 1, 1, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 2, 3, 5, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 0, 1, 1, 4, 0, 1, 0, 1, 0, 1, 0, 6, 0, 1, 2, 1, 0, 1, 0, 2, 4, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 4, 0, 7
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OFFSET
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1,8
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COMMENTS
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Number of solutions, (s,t,k), to s^2 + t^2 = k*n such that s + t = n, 1 <= s <= t and 1 <= k <= n-1. - Wesley Ivan Hurt, Oct 01 2020
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LINKS
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FORMULA
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a(n) = Sum_{i=1..floor(n/2)} (1 - ceiling(2*i*(n-i)/n) + floor(2*i*(n-i)/n)).
a(n) = Sum_{i=1..floor(n/2)} Sum_{k=1..n-1} [i^2 + (n-i)^2 = n*k], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Oct 01 2020
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MATHEMATICA
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Table[Sum[1 - Ceiling[2*i*(n - i)/n] + Floor[2*i*(n - i)/n], {i, Floor[n/2]}], {n, 100}]
Table[Sum[Sum[KroneckerDelta[(i^2 + (n - i)^2)/k, n], {k, n - 1}], {i, Floor[n/2]}], {n, 100}]
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PROG
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(PARI) A337101(n) = { my(u, t); sum(s=1, n\2, t = n-s; u = (s^2 + t^2); (!(u%n) && (u/n) <= n-1)); }; \\ Antti Karttunen, Dec 12 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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