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%I #29 Sep 17 2019 16:54:57
%S 0,0,0,0,1,0,1,0,1,0,1,0,2,1,0,0,1,0,3,2,1,0,4,0,3,2,1,0,6,0,5,4,3,2,
%T 1,0,7,6,5,0,4,0,3,2,1,0,1,0,3,2,1,0,3,3,2,2,1,0,16,0,15,14,13,12,11,
%U 0,10,9,8,0,7,0,6,5,4,3,2,0,1,0,1,0,13,13
%N Let n=r_1*r_2*...*r_k is Fermi-Dirac factorization of n (see comment). Set g(n) = n + k - 1 and g_i, i>=0 (g_0(n) = n, g_1=g), is i-th iteration of g. a(n) is the minimal i such that g_i(n) is in A050376.
%C Recall that every n>=2 has a unique factorization over distinct numbers from A050376 which is called Fermi-Dirac factorization of n. The sequence is a dual to A213980.
%C Conjecture: a(n) exists for every n >= 2.
%H Amiram Eldar, <a href="/A214625/b214625.txt">Table of n, a(n) for n = 2..10000</a>
%e Since 24 = 2*3*4, then g_1(24) = 24 + 3 - 1 = 26; analogously, g_1(26) = 26 +2 -1 = 27, g_1(27) = 27 + 2 - 1 = 28, g_1(28) = 28 + 2 - 1 = 29 is in A050376. We used 4 iterations, therefore, a(24) = 4.
%t f[1]=0; f[n_] := Plus @@ (DigitCount[Last /@ FactorInteger[n], 2, 1]); g[n_] := n + f[n] - 1; a[n_] := Length @ FixedPointList[g, n]; Array[a, 30] (* _Amiram Eldar_, Sep 17 2019 *)
%Y Cf. A050376, A213980.
%K nonn
%O 2,13
%A _Vladimir Shevelev_, Feb 16 2013
%E a(63) corrected by _Amiram Eldar_, Sep 17 2019
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