A083546 -id:A083546 - cp's OEIS frontend

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.

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

A083545 Numbers k such that the geometric mean of the Euler totient function of k and k+1 is an integer.

Original entry on oeis.org

1, 3, 15, 19, 95, 104, 125, 164, 194, 255, 259, 341, 491, 495, 504, 512, 513, 584, 591, 629, 679, 755, 775, 975, 1024, 1147, 1247, 1254, 1260, 1313, 1358, 1463, 1469, 1538, 1615, 1728, 1919, 1962, 1970, 2047, 2071, 2090, 2204, 2299, 2321, 2345, 2404, 2625

Offset: 1

Labos Elemer, May 21 2003

nonn

			19 is a term since phi(19) = 18, phi(20) = 8, 8*18 = 144 = 12^2.

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

Cf. A000010, A083542, A066813, A058515, A083538, A083542, A083546.

f[x_] := EulerPhi[x]; Do[s=Sqrt[f[n+1]*f[n]]; If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Position[Partition[EulerPhi[Range[2700]],2,1],?(IntegerQ[GeometricMean[ #]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 13 2020 *)

a(n) = x is such that sqrt(A000010(x)*A000010(x+1)) is an integer. Values of solutions x to phi(x) * phi(x+1) = A083542(x) = y^2.

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

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.

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A083545 Numbers k such that the geometric mean of the Euler totient function of k and k+1 is an integer.

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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.

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