A058515 -id:A058515 - cp's OEIS frontend

A049559 a(n) = gcd(n - 1, phi(n)).

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 22, 1, 4, 1, 2, 3, 28, 1, 30, 1, 4, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 6, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 1, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 18, 1, 4

Offset: 1

Labos Elemer, Dec 28 2000

nonn

For prime n, a(n) = n - 1. Question: do nonprimes exist with this property?
Answer: No. If n is composite then a(n) < n - 1. - Charles R Greathouse IV, Dec 09 2013
Lehmer's totient problem (1932): are there composite numbers n such that a(n) = phi(n)? - Thomas Ordowski, Nov 08 2015
a(n) = 1 for n in A209211. - Robert Israel, Nov 09 2015

			a(9) = 2 because phi(9) = 6 and gcd(8, 6) = 2.
a(10) = 1 because phi(10) = 4 and gcd(9, 4) = 1.

Richard K. Guy, Unsolved Problems in Number Theory, B37.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lehmer's Totient Problem

Cf. A000010, A002322, A039766, A063994, A160595, A209211, A219428, A264012, A264024, A280262, A283656, A283872, A284089, A284440, A318827, A318829, A318830, A330747 (ordinal transform), A340195.

Cf. also A009195, A058515, A058663, A187730, A258409, A339964, A340071, A340081, A340087 for more or less analogous sequences.

[Gcd(n-1, EulerPhi(n)): n in [1..80]]; // Vincenzo Librandi, Oct 13 2018

seq(igcd(n-1, numtheory:-phi(n)), n=1..100); # Robert Israel, Nov 09 2015

Table[GCD[n - 1, EulerPhi[n]], {n, 93}] (* Michael De Vlieger, Nov 09 2015 *)

a(n)=gcd(eulerphi(n),n-1) \\ Charles R Greathouse IV, Dec 09 2013

from sympy import totient, gcd
print([gcd(totient(n), n - 1) for n in range(1, 101)]) # Indranil Ghosh, Mar 27 2017

a(p^m) = a(p) = p - 1 for prime p and m > 0. - Thomas Ordowski, Dec 10 2013
From Antti Karttunen, Sep 09 2018: (Start)
a(n) = A000010(n) / A160595(n) = A063994(n) / A318829(n).
a(n) = n - A318827(n) = A000010(n) - A318830(n).
(End)
a(n) = gcd(A000010(n), A219428(n)) = gcd(A000010(n), A318830(n)). - Antti Karttunen, Jan 05 2021

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

Original entry on oeis.org

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

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

A049586 a(n) is the GCD of the cototients (A051953) of n and n+1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3

Offset: 0

Labos Elemer, Dec 28 2000

nonn

Most of the terms are 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000

Cf. A000010, A051953, A058515.

[GCD((n+1-EulerPhi(n+1)), n-EulerPhi(n)): n in [1..100]]; // Vincenzo Librandi, Aug 10 2017

Map[GCD @@ Map[# - EulerPhi@ # &, #] &, Partition[Range@ 106, 2, 1]] (* Michael De Vlieger, Aug 09 2017 *)

a(n) = gcd(n + 1 - phi(n+1), n - phi(n)).

a(n) = gcd(A051953(n+1), A051953(n)).

A066813 a(n) = lcm(phi(n), phi(n+1)).

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 12, 12, 12, 20, 20, 12, 12, 24, 8, 16, 48, 18, 72, 24, 60, 110, 88, 40, 60, 36, 36, 84, 56, 120, 240, 80, 80, 48, 24, 36, 36, 72, 48, 80, 120, 84, 420, 120, 264, 506, 368, 336, 420, 160, 96, 312, 468, 360, 120, 72, 252, 812, 464, 240, 60, 180, 288, 96

Offset: 1

Benoit Cloitre, Jan 20 2002

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000010 (phi), A058515, A083542.

LCM@@EulerPhi[#]&/@Partition[Range[100],2,1] (* Harvey P. Dale, May 07 2011 *)

a(n) = { lcm(eulerphi(n), eulerphi(n+1)) } \\ Harry J. Smith, Mar 29 2010

a(n) = A083542(n)/A058515(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).

A063444 Smallest number such that GCD of EulerPhi of 2 consecutive integer equals 2n.

Original entry on oeis.org

3, 12, 13, 15, 121, 35, 86, 64, 37, 99, 726, 72, 158, 196, 61, 96, 4931, 73, 7639, 175, 343, 267, 2302, 104, 250, 676, 162, 637, 3481, 154, 21142, 192, 2178, 411, 5041, 814, 446, 1145, 157, 164, 6971, 1348, 14878, 1334, 542, 2115, 22090, 193, 2842, 2200

Offset: 1

Labos Elemer, Jul 24 2001

nonn

			n = 10, a(10) = 99, Phi(99) = 60, Phi(100) = 40, GCD[60,40] = 2n = 20.

Harry J. Smith, Table of n, a(n) for n=1,...,350

A000010, A058515.

{ for (n=1, 350, x=1; while (gcd(eulerphi(x+1), eulerphi(x)) != 2*n, x++); write("b063444.txt", n, " ", x) ) } \\ Harry J. Smith, Aug 21 2009

Min{x; GCD[Phi[x+1], Phi[x]]=2n}=Min{x; A058515[x]=2n}

A058656 a(n) = gcd(n+1, phi(n)).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 8, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 3, 8, 1, 2, 1, 2, 5, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 2, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 3, 8, 1, 2, 1, 4, 1, 2, 1, 24, 1, 2, 3, 20, 1, 2, 1

Offset: 1

Labos Elemer, Dec 28 2000

nonn

Compare sequences gcd(x, phi(n)), where x = n-1, n, or n+1.

			For n = 12, 13, 14, 15: n+1 = 13, 14, 15, 16; phi(n) = 4, 12, 12, 8; a(n) = gcd(13,4), gcd(14,12), gcd(15,12), gcd(16,8) = 1, 2, 3, 8, respectively.

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

Cf. A000010, A009195, A058515.

Table[GCD[n+1,EulerPhi[n]],{n,110}] (* Harvey P. Dale, Nov 17 2011 *)

a(n) = gcd(n+1, eulerphi(n)); \\ Amiram Eldar, Mar 13 2025

Offset corrected by Sean A. Irvine, Aug 11 2022

A070010 GCD of consecutive values of sum-of-proper divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 11, 1, 1, 1, 6, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 23, 1, 1, 1, 1, 3, 1, 1, 5, 25, 1, 1, 1, 1, 3, 1, 1, 1, 1

Offset: 1

Labos Elemer, Apr 11 2002

nonn

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

Cf. A000203, A001065; GCD of various consecutive function values: A048586, A057467, A058515, A060778, A060780, A069896.

GCD[DivisorSigma[1, n+1]-(n+1), DivisorSigma[1, n]-n]

A001065(n) = (sigma(n) - n);
A070010(n) = gcd(A001065(n),A001065(1+n)); \\ Antti Karttunen, Oct 05 2017

a(n) = gcd(A001065(n+1), A001065(n)).

cp's OEIS Frontend

A049559 a(n) = gcd(n - 1, phi(n)).

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

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

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A049586 a(n) is the GCD of the cototients (A051953) of n and n+1.

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A066813 a(n) = lcm(phi(n), phi(n+1)).

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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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