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A337945
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Numbers m with a solution (s,t,k) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.
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5
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2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112, 114, 116, 117, 118, 120
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OFFSET
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1,1
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LINKS
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FORMULA
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k is a term <=> Sum_{i=1..floor(k/2)} Sum_{j=1..k-1} KroneckerDelta((i^2 + (k - i)^2)/j, k) > 0.
Conjecture: k is a term <=> k * Clausen(k, 1) <> 2 * Clausen(k, 0), (Clausen = A160014). In other words: k is in this sequence iff it is not an odd squarefree number. - Peter Luschny, Jun 08 2023
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EXAMPLE
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8 is in the sequence since it has the solutions (s,t,k) = (4,4,4) and (2,6,5) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.
9 is in the sequence since it has the solution (s,t,k) = (3,6,5) such that s^2 + t^2 = k*m, s + t = m, 1 <= s <= t and 1 <= k <= m - 1.
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MAPLE
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# Quite inefficient compared to the conjectured formula.
KD := (n, k) -> Physics:-KroneckerDelta[n, k]:
S := k -> local i, j; add(add(KD((i^2 + (k - i)^2)/j , k), j = 1..k-1),
i = 1..floor(k/2)): select(k -> S(k) > 0, [seq(k, k = 1..40)]); # Peter Luschny, Jun 08 2023
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MATHEMATICA
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Table[If[Sum[Sum[KroneckerDelta[(i^2 + (n - i)^2)/k, n], {k, n - 1}], {i, Floor[n/2]}] > 0, n, {}], {n, 120}] // Flatten
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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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