A333820 - OEIS

%I #25 Apr 11 2020 06:28:46

%S 3,6,12,22,31,36,46,47,52,69,67,93,90,117,90,119,93,146,98,166,135,

%T 195,117,242,133,236,156,258,139,278,137,306,204,280,158,367,161,348,

%U 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

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

%C 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.

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

%D 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 Eric W. Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach's Conjecture.</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%e 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}.

%t 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]

%Y Cf. A306513, A306513, A333819.

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Apr 06 2020