

A306513


The number of unordered pairs of coprime integers q and r such that phi(q) + phi(r) = 2n.


1



1, 1, 5, 7, 12, 10, 19, 18, 20, 21, 35, 32, 39, 42, 38, 37, 48, 46, 45, 58, 64, 63, 69, 73, 58, 93, 71, 70, 81, 92, 72, 113, 96, 94, 90, 100, 79, 158, 120, 95, 131, 153, 84, 147, 129, 132, 126, 150, 92, 179, 157, 150, 149, 187, 92, 224, 177, 166, 173, 207, 124
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OFFSET

1,3


COMMENTS

Paul ErdÅ‘s and Leo Moser conjectured that, for any even number 2n, there exist integers q and r such that phi(q) + phi(r) = 2n with gcd(q, r) = 1. Adding to this conjecture the requirement that q and r be prime yields the Goldbach Conjecture. The replacement of the requirement that q and r be prime with the relaxed requirement that they be coprime was done in an effort to solve the Goldbach Conjecture.


REFERENCES

George E. Andrews, Number Theory, Chapter 6, Arithmetic Functions, Section 61, Combinatorial Study of Phi(n), pp. 7582, Dover Publishing, NY, 1971.


LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..1150
Eric W. Weisstein's World of Mathematics, Goldbach's Conjecture.
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture


EXAMPLE

a(1) = 1 with {q, r} = {1,2};
a(2) = 1 with {q, r} = {3,4};
a(3) = 5 because phi(q) + phi(r) = 6 for the pairs {q, r} = {3,5}, {3,8}, {3,10}, {4,5} & {5,6}; etc.


MATHEMATICA

f[n_] := Block[{c = 0, q = 1}, While[q < 12n, epq = EulerPhi[q]; r = 12n + 125; While[r > q, If[ GCD[q, r] == 1 && epq + EulerPhi[r] == 2 n, c++]; r]; q++]; c]; Array[f, 61]


CROSSREFS

Cf. A000010, A005277, A079695, A002375.
Sequence in context: A082565 A309695 A086255 * A286901 A171490 A047382
Adjacent sequences: A306510 A306511 A306512 * A306514 A306515 A306516


KEYWORD

nonn


AUTHOR

Robert G. Wilson v, Feb 20 2019


STATUS

approved



