A333820 a(n) is the number of pairs (q,r) such that q <= r and phi(q) + phi(r) = 2*n.

3, 6, 12, 22, 31, 36, 46, 47, 52, 69, 67, 93, 90, 117, 90, 119, 93, 146, 98, 166, 135, 195, 117, 242, 133, 236, 156, 258, 139, 278, 137, 306, 204, 280, 158, 367, 161, 348, 230, 372, 226, 443, 168, 452, 280, 364, 207, 555, 195, 443, 294, 553, 237, 556, 177, 637, 326, 473, 275, 770, 225, 553, 322, 660, 283, 759, 213, 755, 364, 572

Offset: 1

Author

Robert G. Wilson v, Apr 06 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. Therefore, they conjecture a(n) > 0 for all ns.

Examples

			a(2) = 6 because for the pairs {q, r} the following pairs when phi(q) + phi(r) = 4; {3,3}, {3,4}, {3,6}, {4,4}, {4,6}, {6,6}.
		

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. A306513, A306513, A333819.

Programs

Mathematica

f[n_] := Block[{c = 0, q = 1}, While[q < 12n, epq = EulerPhi[q]; r = 12n + 25; While[r >= q, If[ epq + EulerPhi[r] == 2 n, c++; AppendTo[lst, {q, r}]]; r--]; q++]; c]; Array[f, 60]