A102641 Compute the greatest prime factors (GPFs, A006530()) of -j + 2^n for j = 0, 1, ..., L. a(n) is the maximal length L of such a sequence in which the greatest prime factors are increasing with decreasing j.

1, 2, 2, 4, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 3, 2, 4, 4, 4, 2, 3, 4, 4, 2, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 6, 2, 4, 2, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 6, 4, 2, 3, 2, 3, 4, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 4, 2, 4, 4, 4, 2, 3, 3, 4, 2, 4, 2, 3, 5

Offset: 1

Author

Labos Elemer, Jan 21 2005

Keywords

• nonn

Comments

A006530(2^n)=2 is a local minimum. Going either upward or downward with the argument, the largest prime factors are increasing for a while. Here the maximal length of increasing greatest-prime-factor sequences are given when going downward with the arguments. Compare with A102640.

Examples

			For n = 12: 2^10 = 4096. The greatest prime factors of 4096, 4095, 4094, 4093 are as follows: {2, 13, 89, 4093}. A006530(4092) = 31 is already smaller than A006530(4093). Thus the length of the increasing GPF sequence is 4 = a(12).
		

Crossrefs

Cf. A006530, A102640, A102642, A102643, A102644.

Programs

Mathematica

With[{nn = 12}, Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Table[FactorInteger[m][[-1, 1]], {m, k, k - nn, -1}]][2^n], {n, 105}] ] (* Michael De Vlieger, Jul 24 2017 *)