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%I #82 Jul 19 2024 00:30:19
%S 1,1,2,3,5,11,26,68,160,226,198,576,1246,122,328,5470,3048,22554,
%T 69648,24184,262752,4386396,70190,22222,4639830,1914046,3227106,
%U 35917950,2738592,325870,124850,8375086,7397758,129192,6948110,10496178,82166,740450,44446188,1473852,184450,342342,14496,740,2906,7040,53772,20464
%N a(n) = gpf(a(n-3))*gpf(a(n-2)) + gpf(a(n-1)), with a(1)=a(2)=1 and a(3)=2 and where gpf(n) is the greatest prime dividing n, A006530(n).
%C The sequence appears to oscillate about its mean, with greater amplitude each time (an empirical observation).
%C The sequence seems to always give a value for a(n) that is relatively prime to each of g(n-1), g(n-2), and g(n-3), where g(i) = gpf(a(i)) (empirical observation).
%C Proof of the second observation: If a(n) is relatively prime to g(n-1), g(n-2), and g(n-3), then so is g(n). In particular, if g(n) is relatively prime to g(n-1) and g(n-2), then g(n-2)*g(n-1) + g(n) = a(n+1) is relatively prime to g(i) for i=n-2,n-1,n. - _Danny Rorabaugh_, Dec 01 2015
%H Sean A. Irvine, <a href="/A258804/b258804.txt">Table of n, a(n) for n = 1..87</a> (terms 1..80 from Robert Israel)
%e Let gpf(x) = A006530(x).
%e a(4) = gpf(a(1))*gpf(a(2)) + gpf(a(3)) = 1*1 + 2 = 3.
%e a(5) = gpf(a(2))*gpf(a(3)) + gpf(a(4)) = 1*2 + 3 = 5.
%p gpf:= x -> max(numtheory:-factorset(x)):
%p gpf(1):= 1:
%p a:= proc(n) option remember;
%p gpf(procname(n-3))*gpf(procname(n-2))+gpf(procname(n-1))
%p end proc:
%p a(1):= 1: a(2):= 1: a(3):=2:
%p seq(a(n),n=1..70); # _Robert Israel_, Dec 01 2015
%t gpf[n_] := FactorInteger[n][[-1, 1]];
%t a[n_] := a[n] = Switch[n, 1, 1, 2, 1, 3, 2, _, gpf[a[n-3]] * gpf[a[n-2]] + gpf[a[n-1]]];
%t Array[a, 70] (* _Jean-François Alcover_, Aug 26 2022 *)
%o (PARI) gpf(n) = if (n==1, 1, vecmax(factor(n)[,1]));
%o lista(nn) = {print1(b = 1, ", "); print1(c = 1, ", "); print1(d = 2, ", "); for (n=1, nn, e = gpf(b)*gpf(c) + gpf(d); print1(e, ", "); b = c; c = d; d = e;);} \\ _Michel Marcus_, Oct 07 2015
%Y Cf. A006530.
%K nonn
%O 1,3
%A _Iago Casabiell González_, Sep 22 2015
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