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

1,2

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

Table of n, a(n) for n=1..105.

Eric W. Weisstein's World of Mathematics, Goldbach's Conjecture.

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

Mathematica

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

Crossrefs

Cf. A000010, A306513, A333820.

Sequence in context: A301415 A210509 A362049 * A216944 A178832 A111233

Adjacent sequences: A333816 A333817 A333818 * A333820 A333821 A333822

Keyword

nonn

Author

Robert G. Wilson v, Apr 06 2020

Status

approved