A214625 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.

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, 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, 0, 10, 9, 8, 0, 7, 0, 6, 5, 4, 3, 2, 0, 1, 0, 1, 0, 13, 13

Offset: 2

Author

Vladimir Shevelev, Feb 16 2013

Keywords

• nonn

Comments

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.

Conjecture: a(n) exists for every n >= 2.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A050376, A213980.

Programs

Mathematica

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 *)

Extensions

a(63) corrected by Amiram Eldar, Sep 17 2019