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A379258 a(n) is the number of iterations of the Euler phi function needed to reach 1 starting at the n-th 3-smooth number.

%I A379258 #11 Dec 20 2024 02:43:54
%S A379258 1,2,3,3,3,4,4,4,5,4,5,5,6,5,6,5,7,6,6,7,6,8,7,6,8,7,7,9,8,7,9,8,7,10,
%T A379258 9,8,8,10,9,8,11,10,9,8,11,10,9,12,9,11,10,9,12,11,10,13,9,12,11,10,
%U A379258 13,10,12,11,14,10,13,12,11,14,10,13,12,15,11,14,11
%N A379258 a(n) is the number of iterations of the Euler phi function needed to reach 1 starting at the n-th 3-smooth number.
%H A379258 Amiram Eldar, <a href="/A379258/b379258.txt">Table of n, a(n) for n = 1..10000</a>
%F A379258 a(n) = A049108(A003586(n)).
%F A379258 a(n) = valuation(A003586(n), 2) + valuation(A003586(n), 3) + 1 + [valuation(A003586(n), 2) == 0] for n > 1, where [] is the Iverson bracket.
%F A379258 a(n) = A022328(n) + A022329(n) + 1 + [n is in A022330], for n > 1.
%F A379258 a(A022330(n)) = n + 2 for n >= 1.
%F A379258 a(A022331(n)) = n + 1 for n >= 0.
%F A379258 a(A202821(n)) = 2*n + 1, for n >= 0.
%e A379258 a(6) = 4 because the 6th 3-smooth number is A003586(6) = 8, and 4 iterations of phi are needed to reach 1: 8 -> 4 -> 2 -> 1.
%t A379258 f[n_] := Module[{e2 = IntegerExponent[n, 2], e3 = IntegerExponent[n, 3]}, e2 + e3 + 1 + Boole[e2 == 0]]; f[1] = 1; With[{max = 3*10^4}, f /@ Sort[Flatten[Table[2^i*3^j, {i, 0, Log2[max]}, {j, 0, Log[3, max/2^i]}]]]]
%o A379258 (PARI) list(lim) = {my(e2, e3); print1(1, ", "); for(k = 2, lim, e2 = valuation(k, 2); e3 = valuation(k, 3); if(k == (1 << e2) * 3^e3, print1(e2 + e3 + 1 + (e2 == 0), ", ")));}
%Y A379258 Cf. A000010, A003586, A049108, A086420, A022328, A022329, A022330, A022331, A202821.
%K A379258 nonn,easy
%O A379258 1,2
%A A379258 _Amiram Eldar_, Dec 19 2024