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%I #16 Jun 28 2021 04:43:19
%S 4,6,24,30,720,168,720,2520,151200,55440,665280,1310400,47520,
%T 454053600,2471040,3598560,4410806400,1769644800,801964800,1150269120,
%U 2283240960
%N Sequence that arises in connection with the number of primes between consecutive squares.
%C Let f(n) = Sum_{p <= n+1} (floor((n+1)^2/p) - floor(n^2/p))log(p) - Sum_{p < q <= n+1} (floor((n+1)^2/(p*q)) - floor(n^2/(p*q)))log(p*q) + Sum_{p < q < r <= n+1} (floor((n+1)^2/(p*q*r)) - floor(n^2/(p*q*r)))*log(p*q*r) - ... The values of the sequence for positive integer n are e^f(n).
%e f(1) = Sum_{p<=2} (floor((n+1)^2/p) - floor(n^2/p))*log(p) = ((4/2) - (1/2))*log(2) = 2*log(2) = log(4).
%e f(2) = Sum_{p<=3} (floor(9/p) - floor(4/p))*log(p) - Sum_{p < q <= 3} (9/(p*q) - 4/(p*q))*log(p*q) = (9/2 - 4/2)*log(2) + (9/3 - 4/3)*log3 - (9/6 - 4/6)*log(6) = 2*log(2) + 2*log(3) - log(6) = log(6).
%e So the first two terms are 4 and 6.
%t (* 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* )
%t 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]}],
%t {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]}],
%t {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]],
%t {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]}],
%t {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]}],
%t {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]] *)
%K nonn,uned
%O 1,1
%A Chris Orr (ckorr2003(AT)yahoo.com), Mar 11 2010
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