A333819 -id:A333819 - cp's OEIS frontend

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

Robert G. Wilson v, Apr 06 2020

nonn

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.

			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}.

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.

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

Cf. A306513, A306513, A333819.

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]

A333838 a(n) is the greatest integer q <= n such that for some r >= q, phi(q) + phi(r) = 2*n.

Original entry on oeis.org

1, 0, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12, 14, 15, 16, 17, 18, 19, 20, 21, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 34, 38, 39, 40, 41, 42, 43, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 53, 55, 56, 57, 58, 59, 60, 60, 61, 62, 64, 65, 66, 64, 68, 68, 70

Offset: 1

Robert G. Wilson v, Apr 07 2020

nonn

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.

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.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric W. Weisstein's World of Mathematics, Goldbach's Conjecture.
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture

Cf. A306513, A306513, A333819, A333820.

f:= proc(n) local q, R;
   for q from n by -1 to 0 do
     R:= numtheory:-invphi(2*n-numtheory:-phi(q));
     if ormap(`>=`,R,q) then return q fi;
   od;
 -1
end proc:
map(f, [$1..100]); # Robert Israel, Sep 15 2024

mbr = Union@Array[EulerPhi@# &, 500]; a[n_] := Block[{q = n}, While[! MemberQ[mbr, 2 n - EulerPhi@q], q--]; q]; Array[a, 70]

Definition corrected by Robert Israel, Sep 15 2024

cp's OEIS Frontend

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

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