cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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).

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Apr 06 2020

Keywords

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

Crossrefs

Cf. A000010, A306513, A333820.

Programs

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

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

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Apr 06 2020

Keywords

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

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]

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

Views

Author

Robert G. Wilson v, Apr 07 2020

Keywords

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.

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

Crossrefs

Cf. A306513, A306513, A333819, A333820.

Programs

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

Extensions

Definition corrected by Robert Israel, Sep 15 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.