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).

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, 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, 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

Offset: 1

Author

Robert G. Wilson v, Apr 06 2020

Keywords

• nonn

Comments

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.

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.

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, ..., .

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

• Eric W. Weisstein's World of Mathematics, Goldbach's Conjecture.
• Wikipedia, Goldbach's conjecture
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A000010, A306513, A333820.

Programs

Mathematica

mbr = Union@ Array[ EulerPhi@# &, 500]; a[n_] := Block[{q = 1}, While[ !MemberQ[mbr, 2n - EulerPhi@ q], q++]; q]; Array[a, 105]