OFFSET
1,4
COMMENTS
It appears that A025527(n) | a(n) for all n >= 1 and A025527(n) = a(n) for infinitely many n. In addition, it seems that a(n)/a(n-1) = A048671(n) for infinitely many n >= 2. However, I have not established these claims.
This sequence appears in formulas for sequences A075827, A075828, A075829, and A075830 (the first one of which was established in 2002 by Michael Somos).
Conjecture: a(n) = n! * Product_{p <= n} p^min(0, v_p(H'(n))), where the product ranges over primes p, H'(n) = Sum_{k=1..n} (-1)^(k+1)/k, and v_p(r) is the p-adic valuation of rational r (checked for n < 1100).
FORMULA
EXAMPLE
MAPLE
b:= proc(n) b(n):= (-(-1)^n/n +`if`(n=1, 0, b(n-1))) end:
a:= n-> (f-> igcd(b(n)*f, f))(n!):
seq(a(n), n=1..30); # Alois P. Heinz, May 18 2020
MATHEMATICA
b[n_] := b[n] = -(-1)^n/n + If[n == 1, 0, b[n-1]];
a[n_] := GCD[b[n] #, #]&[n!];
Array[a, 30] (* Jean-François Alcover, Oct 27 2020, after Alois P. Heinz *)
PROG
(SageMath)
def A():
a, b, n = 1, 1, 2
while True:
yield gcd(a, b)
b, a = a, a + b * n * n
n += 1
a = A(); print([next(a) for _ in range(29)]) # Peter Luschny, May 19 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Petros Hadjicostas, May 17 2020
STATUS
approved