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%I #10 Oct 16 2021 13:05:48
%S 1,2,3,4,4,6,6,8,9,8,8,12,8,12,12,16,12,18,12,16,18,16,16,24,16,16,27,
%T 24,16,24,16,32,24,24,24,36,24,24,24,32,24,36,24,32,36,32,32,48,36,32,
%U 36,32,36,54,32,48,36,32,32,48,32,32,54,64,32,48,32,48,48,48,48,72,48
%N a(n)=b(n, A102442(n)), where b(n,0)=n and b(n,k+1)=A102440(b(n,k)).
%C a(a(n)) = A102440(a(n)) = a(n).
%C Completely multiplicative because A102440 is. The conversion of every prime into a 3-smooth number is independent of any other prime. - _Andrew Howroyd_, Jul 31 2018
%H Andrew Howroyd, <a href="/A102443/b102443.txt">Table of n, a(n) for n = 1..1000</a>
%e See A102442.
%t g[p_] := (* greatest semiprime less than prime p *) g[p] = For[k = p - 1, True, k--, If[PrimeOmega[k] == 2, Return[k]]];
%t A102440[n_] := Product[{p, e} = pe; If[p <= 3, p, g[p]]^e, {pe, FactorInteger[n]}];
%t A102442[n_] := Length[NestWhileList[A102440, n, FactorInteger[#][[-1, 1]] > 3 & ] - 1];
%t a[n_] := b[n, A102442[n]];
%t b[n_, 0] := n;
%t b[n_, k_] := A102440[b[n, k - 1]];
%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Oct 16 2021 *)
%o (PARI) a(n)={while(1, my(f=factor(n)); if(!#select(t->t>3, f[,1]), return(n), n=prod(i=1, #f~, my(p=f[i,1]); while(p>4 && bigomega(p)<>2, p--); p^f[i,2])))} \\ _Andrew Howroyd_, Jul 31 2018
%Y Cf. A003586, A102440, A102442.
%K nonn,mult
%O 1,2
%A _Reinhard Zumkeller_, Jan 09 2005
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