A001837 -id:A001837 - cp's OEIS frontend

A161963 Even numbers n for which phi(n) > phi(n+1).

314, 524, 734, 824, 944, 974, 1154, 1364, 1574, 1754, 1784, 1814, 1994, 2144, 2414, 2474, 2624, 2804, 3044, 3134, 3254, 3314, 3464, 3704, 3884, 4094, 4124, 4304, 4388, 4514, 4724, 4874, 4934, 5114, 5144, 5354, 5444, 5564, 5774, 5864

Offset: 1

David Angell (angell(AT)maths.unsw.edu.au), Jun 22 2009

If n is even then for obvious reasons phi(n) will usually be less than or equal to phi(n+1). These are the first few exceptions.

Observation based upon calculation: More than 95% of the terms of this sequence have the final digit of 4 for n <= 10^7.. - Harvey P. Dale, Jul 24 2012

			314 is in the list because phi(314)=156 and phi(315)=144.

G. C. Greubel, Table of n, a(n) for n = 1..5000 (terms 1..1000 from T. D. Noe)

Cf. A001837, A161962.

[n: n in [1..6000] | (n mod 2 eq 0) and (EulerPhi(n) gt EulerPhi(n+1))]; // G. C. Greubel, Feb 27 2019

Select[2*Range[3000],EulerPhi[#]>EulerPhi[#+1]&] (* Harvey P. Dale, Jul 24 2012 *)

for(n=1, 6000, if(Mod(n,2)==0 && eulerphi(n) > eulerphi(n+1), print1(n", "))) \\ G. C. Greubel, Feb 27 2019

[n for n in (1..6000) if mod(n,2)==0 and euler_phi(n) > euler_phi(n+1)] # G. C. Greubel, Feb 27 2019

a(n) = 2 * A001837(n) (follows from the definition). - Chris Boyd, Mar 15 2014

A001836 Numbers k such that phi(2k-1) < phi(2k), where phi is Euler's totient function A000010.

Original entry on oeis.org

53, 83, 158, 263, 293, 368, 578, 683, 743, 788, 878, 893, 908, 998, 1073, 1103, 1208, 1238, 1268, 1403, 1418, 1502, 1523, 1658, 1733, 1838, 1943, 1964, 2048, 2063, 2153, 2228, 2243, 2258, 2363, 2393, 2423, 2468, 2558, 2573, 2633, 2657, 2678

Offset: 1

N. J. A. Sloane

Jeffrey Shallit, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
V. L. Klee, Jr., Some remarks on Euler's totient function, Amer. Math. Monthly, 54 (1947), 332.
Don Reble, Python program.
J. Shallit, Letter to N. J. A. Sloane, Jul 17 1975.

Cf. A000010, A001837.

a001836 n = a001836_list !! (n-1)
a001836_list = f a000010_list 1 where
   f (u:v:ws) x = if u < v then x : f ws (x + 1) else f ws (x + 1)
-- Reinhard Zumkeller, Jul 11 2014

with(numtheory): A001836:=n->`if`(phi(2*n-1) < phi(2*n), n, NULL): seq(A001836(n), n=1..5*10^3); # Wesley Ivan Hurt, Oct 10 2014

Select[Range[3000], EulerPhi[2# - 1] < EulerPhi[2#] &] (* Harvey P. Dale, Apr 01 2012 *)
Position[Partition[EulerPhi[Range[6000]],2],?(#[[1]]<#[[2]]&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Jul 02 2021 *)

is(n)=eulerphi(2*n-1)Charles R Greathouse IV, Feb 21 2013

from sympy import totient
def ok(n): return totient(2*n - 1) < totient(2*n) # Indranil Ghosh, Apr 29 2017

Corrected and extended by Don Reble, Jan 04 2007

A064804 Numbers 2k such that phi(2k + 2) < phi(2*k).

Original entry on oeis.org

16, 22, 28, 34, 38, 40, 46, 52, 58, 64, 68, 76, 82, 86, 88, 94, 98, 100, 106, 112, 118, 124, 128, 130, 134, 136, 142, 148, 152, 154, 158, 160, 166, 172, 178, 184, 188, 190, 194, 196, 202, 206, 208, 214, 218, 220, 226, 232, 236, 238, 244, 248, 250, 256, 262, 268

Offset: 1

Robert G. Wilson v, Oct 21 2001

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

Cf. A001837.

2Select[ Range[200], EulerPhi[2# + 2] < EulerPhi[2# ] & ]

{ n=e=0; forstep (m=2, 10^9, 2, f=eulerphi(m); if (fHarry J. Smith, Sep 26 2009

a(41)-a(56) from Harry J. Smith, Sep 26 2009

Name clarified by Sean A. Irvine, Jul 28 2023

A064805 Numbers k such that phi(k) > phi(k+2).

Original entry on oeis.org

13, 16, 19, 22, 23, 25, 28, 31, 34, 37, 38, 40, 43, 46, 47, 49, 52, 53, 55, 58, 61, 64, 67, 68, 73, 76, 79, 82, 83, 85, 86, 88, 89, 91, 94, 97, 98, 100, 103, 106, 109, 112, 113, 115, 118, 121, 124, 127, 128, 130, 131, 133, 134, 136, 139, 142, 143, 145, 148, 151, 152

Offset: 1

Robert G. Wilson v, Oct 21 2001

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

Cf. A000010 (phi), A001837.

Select[ Range[150], EulerPhi[ # ] > EulerPhi[ #+2 ] & ]
Position[Partition[EulerPhi[Range[200]],3,1],?(First[#]>Last[#]&), 1, Heads->False]//Flatten (* _Harvey P. Dale, Jun 14 2017 *)

{ n=0; for (m=1, 10^9, f=eulerphi(m); if (eulerphi(m)>eulerphi(m + 2), write("b064805.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Sep 26 2009

a(60)-a(61) from Harry J. Smith, Sep 26 2009

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A161963 Even numbers n for which phi(n) > phi(n+1).

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A001836 Numbers k such that phi(2k-1) < phi(2k), where phi is Euler's totient function A000010.

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A064804 Numbers 2*k such that phi(2*k + 2) < phi(2*k).

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A064805 Numbers k such that phi(k) > phi(k+2).

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A064804 Numbers 2k such that phi(2k + 2) < phi(2*k).