A066813 -id:A066813 - cp's OEIS frontend

A083538 a(n) = sigma(n)*sigma(n+1)/gcd(sigma(n+1), sigma(n))^2.

3, 12, 28, 42, 2, 6, 120, 195, 234, 6, 21, 2, 84, 1, 744, 558, 78, 780, 210, 336, 72, 6, 10, 1860, 1302, 420, 35, 420, 60, 36, 2016, 336, 72, 72, 4368, 3458, 570, 210, 1260, 105, 112, 264, 231, 182, 156, 6, 372, 7068, 589, 744, 1764, 1323, 180, 15, 15, 6, 72, 6, 70

Offset: 1

Labos Elemer, May 21 2003

nonn

a(n) = A060781(n)/A060780(n) = A083539(n)/A060780(n)^2; quotient when lcm(sigma(n+1), sigma(n)) is divided by gcd(sigma(n+1), sigma(n)).

			n=10: sigma(10)=18, sigma(11)=12, lcm(18, 12)=36, gcd(18, 12)=6, a(10) = 36/6 = 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000010, A066813, A058515, A000203, A060781, A060780, A083539.

f[x_] := DivisorSigma[1, x] t=Table[LCM[f[w+1], f[w]]/GCD[f[w+1], f[w]], {w, 1, 128}]
Times@@#/(GCD@@#)^2&/@Partition[DivisorSigma[1,Range[60]],2,1] (* Harvey P. Dale, Feb 17 2016 *)

a(n)=my(x=sigma(n),y=sigma(n+1)); x*y/gcd(x,y)^2 \\ Charles R Greathouse IV, Mar 09 2014

Edited by N. J. A. Sloane, Apr 29 2007

Corrections by Charles R Greathouse IV, Mar 09 2014

A058515 GCD of totients of consecutive integers.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 6, 2, 8, 8, 2, 6, 2, 4, 2, 2, 2, 4, 4, 6, 6, 4, 4, 2, 2, 4, 4, 8, 12, 12, 18, 6, 8, 8, 4, 6, 2, 4, 2, 2, 2, 2, 2, 4, 8, 4, 2, 2, 8, 12, 4, 2, 2, 4, 30, 6, 4, 16, 4, 2, 2, 4, 4, 2, 2, 24, 36, 4, 4, 12, 12, 6, 2, 2, 2, 2, 2, 8, 2, 14, 8, 8, 8, 24, 4, 4, 2, 2, 8, 32, 6

Offset: 1

Labos Elemer, Dec 21 2000

			For n = 61, gcd(phi(62), phi(61)) = gcd(30, 60) = 30, so a(61) = 30.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A049586, A060778, A066813, A083542.

Map[GCD @@ # &, Partition[EulerPhi@ Range@ 98, 2, 1]] (* Michael De Vlieger, Aug 22 2017 *)

a(n) = gcd(eulerphi(n), eulerphi(n+1)); \\ Michel Marcus, Dec 10 2013

a(n) = gcd(phi(n+1), phi(n)), where phi = A000010.

a(n) = A083542(n)/A066813(n). - Amiram Eldar, May 07 2025

Offset corrected to 1 by Michel Marcus, Dec 10 2013

A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.

Original entry on oeis.org

1, 2, 4, 8, 8, 12, 24, 24, 24, 40, 40, 48, 72, 48, 64, 128, 96, 108, 144, 96, 120, 220, 176, 160, 240, 216, 216, 336, 224, 240, 480, 320, 320, 384, 288, 432, 648, 432, 384, 640, 480, 504, 840, 480, 528, 1012, 736, 672, 840, 640, 768, 1248, 936, 720, 960, 864, 1008

Offset: 1

Labos Elemer, May 21 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A002378, A058515, A065474, A066813, A330319 (partial sums).

Cf. A083481, A083539, A092517.

a083542 n = a000010 n * a000010 (n + 1)
a083542_list = zipWith (*) (tail a000010_list) a000010_list
-- Reinhard Zumkeller, Apr 22 2012

a:= n-> (p-> p(n)*p(n+1))(numtheory[phi]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 21 2022

Times @@ EulerPhi@ # & /@ Partition[Range@ 58, 2, 1] (* Michael De Vlieger, Mar 25 2017 *)
Times@@@Partition[EulerPhi[Range[60]],2,1] (* Harvey P. Dale, Oct 29 2019 *)

a(n) = eulerphi(n) * eulerphi(n+1); \\ Amiram Eldar, Jul 10 2024

a(n) = A000010(A002378(n)). - Amiram Eldar, Jul 10 2024
Sum_{k=1..n} a(k) = c * n^3 / 3 + O((n*log(n))^2), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 09 2024
a(n) = A058515(n)*A066813(n). - Amiram Eldar, May 07 2025

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.

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A083538 a(n) = sigma(n)*sigma(n+1)/gcd(sigma(n+1), sigma(n))^2.

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A058515 GCD of totients of consecutive integers.

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A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.

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

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