cp's OEIS Frontend

A290001 Numbers k such that the iteration k -> (sigma(k)+phi(k))/2 terminates at a fraction later than at the first step.

%I A290001 #31 Dec 23 2024 14:53:45
%S A290001 12,14,15,20,24,28,33,34,35,42,48,54,55,56,62,63,69,70,76,77,78,84,85,
%T A290001 86,87,88,90,93,94,95,96,99,102,104,105,108,110,111,112,118,119,120,
%U A290001 122,123,124,125,129,130,132,141,142,143,148,150
%N A290001 Numbers k such that the iteration k -> (sigma(k)+phi(k))/2 terminates at a fraction later than at the first step.
%C A290001 Termination at the first step occurs if k>3 is a square or twice a square, i.e. if k is a term of A028982. So the sequence lists the numbers that end at a fraction but are not of one of these two forms.
%H A290001 Hugo Pfoertner, <a href="/A290001/b290001.txt">Table of n, a(n) for n = 1..101</a>
%H A290001 Hugo Pfoertner, <a href="/A290001/a290001.txt">Table of n, a(n) for n = 1..1136</a> (values above first 101 terms are only guesses)
%H A290001 N. J. A. Sloane, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2017-September/017915.html">Iterating some number-theoretic functions</a>, Posting to Seqfan mailing list, Sep 3, 2017
%H A290001 N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, <a href="https://vimeo.com/237029685">Part I</a>, <a href="https://vimeo.com/237030304">Part 2</a>, <a href="https://oeis.org/A290447/a290447_slides.pdf">Slides.</a> (Mentions this sequence)
%e A290001 42 is in the sequence because the following iteration reaches an odd sigma(k):
%e A290001    k        sigma(k)     phi(k)
%e A290001    42          96          12  k->(sigma(k)+phi(k))/2, (96+12)/2=54
%e A290001    54         120          18  k->(sigma(k)+phi(k))/2, (120+18)/2=69
%e A290001    69          96          44  ...
%e A290001    70         144          24  ...
%e A290001    84         224          24
%e A290001   124         224          60
%e A290001   142         216          70
%e A290001   143         168         120  k->(sigma(k)+phi(k))/2, (168+120)/2=144
%e A290001   144         403          48  sigma(144) is odd, iteration terminates
%Y A290001 Cf. A000010, A000203, A028982, A289997, A291790, A291791.
%K A290001 nonn
%O A290001 1,1
%A A290001 _Hugo Pfoertner_, Sep 03 2017
%E A290001 b-file curtailed beyond last certain value, moved other values to a-file. - _N. J. A. Sloane_, Oct 01 2017