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A333838 a(n) is the greatest integer q <= n such that for some r >= q, phi(q) + phi(r) = 2*n.

%I A333838 #18 Feb 16 2025 08:33:59
%S A333838 1,0,3,4,5,6,7,8,9,10,11,12,12,14,15,16,17,18,19,20,21,21,23,24,25,26,
%T A333838 27,28,29,30,31,32,33,34,35,36,34,38,39,40,41,42,43,43,45,46,47,48,49,
%U A333838 50,51,52,53,53,55,56,57,58,59,60,60,61,62,64,65,66,64,68,68,70
%N A333838 a(n) is the greatest integer q <= n such that for some r >= q, phi(q) + phi(r) = 2*n.
%C A333838 Paul Erdős and Leo Moser conjectured that, for any even numbers 2*n, there exist integers q and r such that phi(q) + phi(r) = 2*n.
%D A333838 George E. Andrews, Number Theory, Chapter 6, Arithmetic Functions, 6-1 Combinatorial Study of Phi(n) page 75-82, Dover Publishing, NY, 1971.
%D A333838 Daniel Zwillinger, Editor-in-Chief, CRC Standard Mathematical Tables and Formulae, 31st Edition, 2.4.15 Euler Totient pages 128-130, Chapman & Hall/CRC, Boca Raton, 2003.
%H A333838 Robert Israel, <a href="/A333838/b333838.txt">Table of n, a(n) for n = 1..10000</a>
%H A333838 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach's Conjecture.</a>
%H A333838 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A333838 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%p A333838 f:= proc(n) local q, R;
%p A333838    for q from n by -1 to 0 do
%p A333838      R:= numtheory:-invphi(2*n-numtheory:-phi(q));
%p A333838      if ormap(`>=`,R,q) then return q fi;
%p A333838    od;
%p A333838  -1
%p A333838 end proc:
%p A333838 map(f, [$1..100]); # _Robert Israel_, Sep 15 2024
%t A333838 mbr = Union@Array[EulerPhi@# &, 500]; a[n_] := Block[{q = n}, While[! MemberQ[mbr, 2 n - EulerPhi@q], q--]; q]; Array[a, 70]
%Y A333838 Cf. A306513, A306513, A333819, A333820.
%K A333838 nonn
%O A333838 1,3
%A A333838 _Robert G. Wilson v_, Apr 07 2020
%E A333838 Definition corrected by _Robert Israel_, Sep 15 2024