A337945 - OEIS

%I #20 Jun 08 2023 18:04:45

%S 2,4,6,8,9,10,12,14,16,18,20,22,24,25,26,27,28,30,32,34,36,38,40,42,

%T 44,45,46,48,49,50,52,54,56,58,60,62,63,64,66,68,70,72,74,75,76,78,80,

%U 81,82,84,86,88,90,92,94,96,98,99,100,102,104,106,108,110,112,114,116,117,118,120

%N 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.

%F k is a term <=> Sum_{i=1..floor(k/2)} Sum_{j=1..k-1} KroneckerDelta((i^2 + (k - i)^2)/j, k) > 0.

%F 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

%e 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.

%e 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.

%p # Quite inefficient compared to the conjectured formula.

%p KD := (n, k) -> Physics:-KroneckerDelta[n, k]:

%p S := k -> local i, j; add(add(KD((i^2 + (k - i)^2)/j , k), j = 1..k-1),

%p i = 1..floor(k/2)): select(k -> S(k) > 0, [seq(k, k = 1..40)]); # _Peter Luschny_, Jun 08 2023

%t Table[If[Sum[Sum[KroneckerDelta[(i^2 + (n - i)^2)/k, n], {k, n - 1}], {i, Floor[n/2]}] > 0, n, {}], {n, 120}] // Flatten

%Y Cf. A337101, A160014, A039956.

%K nonn

%O 1,1

%A _Wesley Ivan Hurt_, Oct 01 2020