A083545 -id:A083545 - cp's OEIS frontend

A083546 The geometric mean of the Euler totient function of 2 consecutive integers {k, k+1} when it is an integer.

1, 2, 8, 12, 48, 48, 60, 80, 96, 128, 144, 180, 280, 240, 240, 288, 288, 288, 336, 288, 384, 360, 480, 480, 640, 720, 672, 600, 576, 720, 720, 720, 672, 864, 960, 864, 960, 1080, 1008, 1408, 1296, 960, 1008, 1320, 1260, 1056, 1440, 1200, 1728, 1440, 1296

Offset: 1

Labos Elemer, May 21 2003

nonn

			12 is a term since sqrt(phi(19) * phi(20)) = sqrt(18 * 8) = sqrt(144) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A083542, A083542, A066813, A058515, A083542, A083545.

f[x_] := EulerPhi[x]; Do[s=Sqrt[f[n+1]*f[n]]; If[IntegerQ[s], Print[s]], {n, 1, 5000}]
Select[Sqrt[#]&/@(Times@@@Partition[EulerPhi[Range[3000]],2,1]),IntegerQ] (* Harvey P. Dale, Nov 04 2020 *)

a(n) = sqrt(A000010(x) * A000010(x+1)) = sqrt(phi(x) * phi(x+1)) = sqrt(A083542(x)) where x = A083545(n).

A083547 a(n) = sqrt(sqrt(phi(A083546(n)) * phi(1+A083546(n)))), the 4th root of product of totients of terms and 1+terms of A082788.

Original entry on oeis.org

1, 12, 24, 36, 36, 60, 60, 72, 80, 96, 120, 120, 120, 144, 144, 168, 180, 240, 264, 360, 360, 432, 480, 504, 480, 480, 720, 720, 720, 720, 840, 840, 864, 840, 840, 840, 840, 960, 900, 960, 960, 1080, 1260, 1224, 1320, 1320, 1440, 1440, 1320, 1440, 1440, 1728

Offset: 1

Labos Elemer, May 21 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A000010, A082788, A083538, A083542, A083545, A083546.

f[x_] := EulerPhi[x]; Do[s=Sqrt[Sqrt[f[n+1]*f[n]]]; If[IntegerQ[s], Print[s]], {n, 1, 1000000}]

a(29)-a(52) from Amiram Eldar, Apr 09 2021

A083549 Quotient if least common multiple (lcm) of cototient values of consecutive integers is divided by the greatest common divisor (gcd) of the same pair of consecutive numbers.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 12, 2, 6, 8, 8, 8, 56, 56, 8, 12, 12, 12, 12, 12, 12, 16, 80, 70, 126, 144, 16, 22, 22, 16, 208, 234, 198, 264, 24, 20, 12, 40, 24, 30, 30, 24, 56, 56, 24, 32, 224, 210, 570, 532, 28, 36, 60, 480, 672, 70, 30, 44, 44, 32, 864, 864, 544, 782, 46, 36, 900

Offset: 1

Labos Elemer, May 22 2003

nonn

			n=33: cototient(33) = 33-20 = 13, cototient(34) = 34-16 = 18;
lcm(13,18) = 234, gcd(13,18) = 1, so a(34) = 234.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A051953, A083538, A083539, A083540, A083541, A083542, A083543, A083544, A083545, A083546, A083547, A083548, A083549, A083550, A083551, A083552, A083553, A083554, A083555, A049586.

f[x_] := x-EulerPhi[x]; Table[LCM[f[w+1], f[w]]/GCD[f[w+1], f[w]], {w, 69}]
(* Second program: *)
Map[Apply[LCM, #]/Apply[GCD, #] &@ Map[# - EulerPhi@ # &, #] &, Partition[Range[69], 2, 1]] (* Michael De Vlieger, Mar 17 2018 *)

a(n) = lcm(A051953(n), A051952(n+1))/gcd(A051953(n), A051952(n+1)) = lcm(cototient(n+1), cototient(n))/A049586(n).

cp's OEIS Frontend

A083546 The geometric mean of the Euler totient function of 2 consecutive integers {k, k+1} when it is an integer.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A083547 a(n) = sqrt(sqrt(phi(A083546(n)) * phi(1+A083546(n)))), the 4th root of product of totients of terms and 1+terms of A082788.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A083549 Quotient if least common multiple (lcm) of cototient values of consecutive integers is divided by the greatest common divisor (gcd) of the same pair of consecutive numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula