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

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, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 34, 38, 39, 40, 41, 42, 43, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 53, 55, 56, 57, 58, 59, 60, 60, 61, 62, 64, 65, 66, 64, 68, 68, 70

Offset: 1

Author

Robert G. Wilson v, Apr 07 2020

Keywords

• nonn

Comments

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.

References

• George E. Andrews, Number Theory, Chapter 6, Arithmetic Functions, 6-1 Combinatorial Study of Phi(n) page 75-82, Dover Publishing, NY, 1971.
• 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.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric W. Weisstein's World of Mathematics, Goldbach's Conjecture.
• Wikipedia, Goldbach's conjecture
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A306513, A306513, A333819, A333820.

Programs

Maple

f:= proc(n) local q, R;
   for q from n by -1 to 0 do
     R:= numtheory:-invphi(2*n-numtheory:-phi(q));
     if ormap(`>=`,R,q) then return q fi;
   od;
 -1
end proc:
map(f, [$1..100]); # Robert Israel, Sep 15 2024

Mathematica

mbr = Union@Array[EulerPhi@# &, 500]; a[n_] := Block[{q = n}, While[! MemberQ[mbr, 2 n - EulerPhi@q], q--]; q]; Array[a, 70]

Extensions

Definition corrected by Robert Israel, Sep 15 2024