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A333819 a(n) is the least integer q > 0 such that for some integer r, phi(q) + phi(r) = 2*n; where phi(n) is Euler's totient function (A000010).

%I A333819 #20 Feb 16 2025 08:33:59
%S A333819 1,3,3,3,3,3,3,5,3,3,3,3,3,5,3,3,3,5,3,5,3,3,3,3,3,5,3,3,3,3,3,5,3,3,
%T A333819 5,3,3,5,7,3,3,3,3,5,3,5,3,5,3,5,3,3,3,3,3,3,3,5,3,5,3,5,7,3,3,3,3,5,
%U A333819 3,3,3,5,3,5,3,3,5,7,3,5,3,3,3,3,3,5,3,5,3,3,3,5,3,5,7,3,3,5,3,3,3,5,3,5,3
%N A333819 a(n) is the least integer q > 0 such that for some integer r, phi(q) + phi(r) = 2*n; where phi(n) is Euler's totient function (A000010).
%C A333819 Paul Erdös 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.
%C A333819 The only time phi is odd, it equals 1. Therefore, the only time that phi(q) + phi(r) = 2*n-1 (for n>0) has no solution is when 2*n-2 is a member of A005277 = 2*A079695.
%C A333819 The first occurrence of 2*k-1, or 0 if not possible, is k=1,2,3,...: 1, 2, 8, 39, 0, 124, 204, 208, 2024, 3473, 0, 2983, 2023, ..., .
%D A333819 George E. Andrews, Number Theory, Chapter 6, Arithmetic Functions, 6-1 Combinatorial Study of Phi(n) page 75-82, Dover Publishing, NY, 1971.
%D A333819 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 A333819 Eric W. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach's Conjecture.</a>
%H A333819 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A333819 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%t A333819 mbr = Union@ Array[ EulerPhi@# &, 500]; a[n_] := Block[{q = 1}, While[ !MemberQ[mbr, 2n - EulerPhi@ q], q++]; q]; Array[a, 105]
%Y A333819 Cf. A000010, A306513, A333820.
%K A333819 nonn
%O A333819 1,2
%A A333819 _Robert G. Wilson v_, Apr 06 2020