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A334958 GCD of consecutive terms of the factorial times the alternating harmonic series.

%I A334958 #45 Oct 27 2020 08:48:13
%S A334958 1,1,1,2,2,12,12,48,144,1440,1440,17280,17280,241920,18144000,
%T A334958 145152000,145152000,2612736000,2612736000,10450944000,219469824000,
%U A334958 4828336128000,4828336128000,115880067072000,579400335360000,15064408719360000,135579678474240000,26573616980951040000,26573616980951040000
%N A334958 GCD of consecutive terms of the factorial times the alternating harmonic series.
%C A334958 For n = 1..14, we have a(n) = A025527(n), but a(15) = 18144000 <> 3628800 = A025527(15).
%C A334958 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.
%C A334958 This sequence appears in formulas for sequences A075827, A075828, A075829, and A075830 (the first one of which was established in 2002 by _Michael Somos_).
%C A334958 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).
%F A334958 a(n) = gcd(A024167(n+1), A024167(n)) = gcd(A024168(n+1), A024168(n)) = gcd(A024167(n), n!) = gcd(A024168(n), n!) = gcd(A024167(n), A024168(n)).
%e A334958 A024167(4) = 4!*(1 - 1/2 + 1/3 - 1/4) = 14, A024167(5) = 5!*(1 - 1/2 + 1/3 - 1/4 + 1/5) = 94, A024168(4) = 4!*(1/2 - 1/3 + 1/4) = 10, and A024168(5) = 5!*(1/2 - 1/3 + 1/4 - 1/5) = 26. Then a(4) = gcd(14, 94) = gcd(10, 26) = gcd(14, 4!) = gcd(10, 4!) = gcd(14, 10) = 2.
%p A334958 b:= proc(n) b(n):= (-(-1)^n/n +`if`(n=1, 0, b(n-1))) end:
%p A334958 a:= n-> (f-> igcd(b(n)*f, f))(n!):
%p A334958 seq(a(n), n=1..30);  # _Alois P. Heinz_, May 18 2020
%t A334958 b[n_] := b[n] = -(-1)^n/n + If[n == 1, 0, b[n-1]];
%t A334958 a[n_] := GCD[b[n] #, #]&[n!];
%t A334958 Array[a, 30] (* _Jean-François Alcover_, Oct 27 2020, after _Alois P. Heinz_ *)
%o A334958 (SageMath)
%o A334958 def A():
%o A334958     a, b, n = 1, 1, 2
%o A334958     while True:
%o A334958         yield gcd(a, b)
%o A334958         b, a = a, a + b * n * n
%o A334958         n += 1
%o A334958 a = A(); print([next(a) for _ in range(29)]) # _Peter Luschny_, May 19 2020
%Y A334958 Cf. A000142, A024167, A024168, A025527, A048671, A058312, A058313, A075827, A075828, A075829, A075830.
%Y A334958 Cf. A056612 (similar sequence for the harmonic series).
%K A334958 nonn
%O A334958 1,4
%A A334958 _Petros Hadjicostas_, May 17 2020