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MATHEMATICA
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(* The following formula gives up to the first 27 values. It can be extended by including the appropriate number of k-tuple sums to determine more values* )
f[n_] := FullSimplify[ Sum[(Floor[(n + 1)^2/Prime[j]] - Floor[n^2/Prime[j]])*Log[Prime[j]], {j, 1, PrimePi[n + 1]}] - Sum[Sum[(Floor[((n + 1)^2)/(Prime[i]Prime[j])] - Floor[((n)^2)/(Prime[i]Prime[j])])*Log[Prime[i]Prime[j]], {j, i + 1, PrimePi[n + 1]}],
{i, 1, PrimePi[n + 1]}] + Sum[Sum[Sum[(Floor[((n + 1)^2)/(Prime[i]Prime[j]Prime[k])] - Floor[((n)^2)/(Prime[i]Prime[j]Prime[k])])* Log[Prime[i]Prime[j]Prime[k]], {k, j + 1, PrimePi[n + 1]}], {j, i + 1, PrimePi[n + 1]}], {i, 1, PrimePi[n + 1]}] - Sum[Sum[Sum[ Sum[(Floor[((n + 1)^2)/(Prime[i]Prime[j]Prime[k]Prime[q])] - Floor[((n)^2)/(Prime[i]Prime[j]Prime[k]Prime[q])])* Log[Prime[i]Prime[j]Prime[k]Prime[q]], {q, k + 1, PrimePi[n + 1]}], {k, j + 1, PrimePi[n + 1]}], {j, i + 1, PrimePi[n + 1]}],
{i, 1, PrimePi[n + 1]}] + Sum[ Sum[Sum[Sum[ Sum[(Floor[((n + 1)^2)/(Prime[i] Prime[j]Prime[k]Prime[q]Prime[ p])] - Floor[((n)^2)/(Prime[i]Prime[j]Prime[k]Prime[q]Prime[ p])])* Log[Prime[i]Prime[j]Prime[k]Prime[q]Prime[p]], {p, q + 1, PrimePi[n + 1]}], {q, k + 1, PrimePi[n + 1]}], {k, j + 1, PrimePi[n + 1]}], {j, i + 1, PrimePi[n + 1]}], {i, 1, PrimePi[n + 1]}] - Sum[ Sum[Sum[Sum[ Sum[Sum[(Floor[((n + 1)^2)/(Prime[i]Prime[j]Prime[k]Prime[ q]Prime[p]Prime[r])] - Floor[((n)^2)/(Prime[i]Prime[j]Prime[k]Prime[q]Prime[ p]Prime[r])])* Log[Prime[i]Prime[j]Prime[k]Prime[q]Prime[p]Prime[r]],
{r, p + 1, PrimePi[n + 1]}], {p, q + 1, PrimePi[n + 1]}], {q, k + 1, PrimePi[n + 1]}], {k, j + 1, PrimePi[n + 1]}], {j, i + 1, PrimePi[n + 1]}], {i, 1, PrimePi[n + 1]}] + Sum[ Sum[Sum[Sum[ Sum[Sum[Sum[(Floor[((n + 1)^2)/(Prime[i]Prime[j]Prime[k]Prime[ q]Prime[p]Prime[r]Prime[v])] - Floor[((n)^2)/(Prime[i]Prime[j]Prime[k]Prime[ q]Prime[p]Prime[r]Prime[v])])* Log[Prime[i]Prime[j]Prime[k]Prime[q]Prime[p]Prime[ r]Prime[v]], {v, r + 1, PrimePi[n + 1]}], {r, p + 1, PrimePi[n + 1 ]}], {p, q + 1, PrimePi[n + 1]}],
{q, k + 1, PrimePi[n + 1]} ], {k, j + 1, PrimePi[n + 1]}], {j, i + 1, PrimePi[n + 1]}], {i, 1, PrimePi[n + 1]}] - Sum[ Sum[Sum[Sum[ Sum[Sum[Sum[ Sum[(Floor[((n + 1)^2)/(Prime[i]Prime[j]Prime[k]Prime[ q]Prime[p]Prime[r]Prime[v]Prime[u])] - Floor[((n)^2)/(Prime[i]Prime[j]Prime[k]Prime[ q]Prime[p]Prime[r]Prime[v]Prime[u])])* Log[Prime[i]Prime[j]Prime[k]Prime[q]Prime[p]Prime[ r]Prime[v]Prime[u]], {u, v + 1, PrimePi[n + 1]}], {v, r + 1, PrimePi[n + 1]}], {r, p + 1, PrimePi[n + 1]}], {p, q + 1, PrimePi[n + 1]}],
{q, k + 1, PrimePi[n + 1]}], {k, j + 1, PrimePi[n + 1]}], {j, i + 1, PrimePi[n + 1]}], {i, 1, PrimePi[n + 1]}] + Sum[ Sum[Sum[Sum[ Sum[Sum[Sum[ Sum[Sum[(Floor[((n + 1)^2)/(Prime[i]Prime[j]Prime[k]Prime[ q]Prime[p]Prime[r]Prime[v]Prime[ u]Prime[a])] - Floor[((n)^2)/(Prime[i]Prime[j]Prime[k]Prime[ q]Prime[p]Prime[r]Prime[v]Prime[ u]Prime[a])])* Log[Prime[i]Prime[j]Prime[k]Prime[q]Prime[p]Prime[ r]Prime[v]Prime[u]Prime[a]], {a, u + 1, Prime[n + 1]}], {u, v + 1, PrimePi[n + 1]}], {v, r + 1, PrimePi[n + 1]}], {r, p + 1, PrimePi[n + 1]}], {p, q + 1, PrimePi[n + 1]}], {q, k + 1, PrimePi[n + 1]}], {k, j + 1, PrimePi[n + 1]}], {j, i + 1, PrimePi[n + 1]}], {i, 1, PrimePi[n + 1]}]] (* Then the values are Exp[f[n]] *)
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