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A384452 a(n) is the sum of squares of the unitary divisors of n!.

%I A384452 #29 Jun 09 2025 00:38:05
%S A384452 1,5,50,650,16900,547924,27396200,1746641000,139773881000,
%T A384452 13460683752200,1642203417768400,236441876606410000,
%U A384452 40195119023089700000,7723888546922636420000,1735183690969722609168800,444206919394766468845892000,128820006624482275965308680000,41737604550102658693597600532800
%N A384452 a(n) is the sum of squares of the unitary divisors of n!.
%H A384452 Project Euler, <a href="https://projecteuler.net/problem=429">Problem 429: Sum of Squares of Unitary Divisors</a>.
%H A384452 Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre%27s_formula">Legendre's formula</a>.
%F A384452 a(n) = Sum_{d|n!} (d^2 if gcd(d,n!//d) = 1).
%F A384452 a(n) = Product_{p <= n, p prime} (p^(2*f(n,p)))+1 with f(n,p) = f(floor(n/p)) + floor(n/p) and f(0,p) = 0 where f(n,p) is equivalent to the Legendre formula.
%F A384452 a(n) = A034676(n!).
%t A384452 f[p_, e_] := p^(2*e)+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 18] (* _Amiram Eldar_, Jun 02 2025 *)
%o A384452 (Python)
%o A384452 from sympy import nextprime
%o A384452 def f(n,p):
%o A384452   if n==0: return 0
%o A384452   return f(n//p,p) + n//p
%o A384452 def a(n):
%o A384452   s,p = 1, 2
%o A384452   while p<=n:
%o A384452     s *= p**(f(n,p)<<1)+1
%o A384452     p = nextprime(p)
%o A384452   return s
%o A384452 print([a(n) for n in range(1, 19)])
%o A384452 (PARI) row(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); } \\ A077610
%o A384452 a(n) = norml2(row(n!)); \\ _Michel Marcus_, Jun 02 2025
%Y A384452 Cf. A000142, A034676, A064028, A077610.
%K A384452 nonn
%O A384452 1,2
%A A384452 _Darío Clavijo_, Jun 02 2025