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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.
4
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
1,2
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.
EXAMPLE
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).
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Jan 21 2005
STATUS
approved