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A102442 Number of iterations needed to transform n by A102440 into a 3-smooth number.

%I A102442 #9 Feb 04 2020 03:58:11
%S A102442 0,0,0,0,1,0,1,0,0,1,2,0,2,1,1,0,2,0,2,1,1,2,3,0,1,2,0,1,3,1,3,0,2,2,
%T A102442 1,0,2,2,2,1,3,1,3,2,1,3,4,0,1,1,2,2,3,0,2,1,2,3,4,1,4,3,1,0,2,2,3,2,
%U A102442 3,1,4,0,4,2,1,2,2,2,3,1,0,3,4,1,2,3,3,2,4,1,2,3,3,4,2,0,3,1,2,1,3,2,3,2,1
%N A102442 Number of iterations needed to transform n by A102440 into a 3-smooth number.
%H A102442 Amiram Eldar, <a href="/A102442/b102442.txt">Table of n, a(n) for n = 1..10000</a>
%F A102442 a(n) = 0 iff n is 3-smooth (A003586);
%F A102442 a(A102444(n)) = n and a(m) < n for m < a(A102444(n)).
%e A102442 A102440(A102440(A102440(41))) = 24 and A102440(24) = 24:
%e A102442 41 -> 3*13 -> 3*[13->2*5] = 2*3*5 -> 2*3*[5->2*2] = 3*2^3 = 24,
%e A102442 therefore a(41) = 3, A102443(41) = 24.
%t A102442 g[p_] := g[p] = For[k = p - 1, True, k--, If[PrimeOmega[k] == 2, Return[k]]]; f[n_] := Product[{p, e} = pe; If[p <= 3, p, g[p]]^e, {pe, FactorInteger[n]}]; a[n_] := -1 + Length @ NestWhileList[f, n, FactorInteger[#][[-1, 1]] > 3 &]; Array[a, 105] (* _Amiram Eldar_, Feb 04 2020 after _Jean-François Alcover_ at A102440 *)
%Y A102442 Cf. A003586, A102440, A102443, A102444.
%K A102442 nonn
%O A102442 1,11
%A A102442 _Reinhard Zumkeller_, Jan 09 2005