A161963 -id:A161963 - cp's OEIS frontend

A161962 Odd numbers k such that phi(k) < phi(k+1).

105, 165, 315, 525, 585, 735, 1155, 1365, 1485, 1575, 1755, 1785, 1815, 1995, 2145, 2205, 2415, 2475, 2535, 2805, 2835, 3003, 3045, 3315, 3465, 3675, 3885, 3927, 4095, 4125, 4305, 4455, 4485, 4515, 4725, 4785, 4845, 4935, 5115, 5145

Offset: 1

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

If k is even then for obvious reasons phi(k) will usually be less than phi(k+1). So it is more interesting to see when this occurs for odd k.

			105 is in the sequence since phi(105)=48 and phi(106)=52.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000010, A001836, A161963.

[n: n in [1..5500] | ((n+1) mod 2 eq 0) and (EulerPhi(n) lt EulerPhi(n+1))]; // G. C. Greubel, Feb 27 2019

with(numtheory): a := proc (n) if `mod`(n, 2) = 1 and phi(n) < phi(n+1) then n else end if end proc: seq(a(n), n = 1 .. 6000); # Emeric Deutsch, Jul 11 2009

Select[Range[5500], OddQ[#] && EulerPhi[#] < EulerPhi[# + 1] &] (* G. C. Greubel, Feb 27 2019 *)
2#-1&/@Flatten[Position[Partition[EulerPhi[Range[5500]],2],?(#[[1]]<#[[2]]&),1,Heads-> False]] (* _Harvey P. Dale, Nov 25 2022 *)

for(n=1, 5500, if(Mod(n+1,2)==0 && eulerphi(n) < eulerphi(n+1), print1(n", "))) \\ G. C. Greubel, Feb 27 2019

[n for n in (1..5500) if mod(n+1,2)==0 and euler_phi(n) < euler_phi(n+1)] # G. C. Greubel, Feb 27 2019

a(n) = 2*A001836(n) - 1. - Jon Maiga / Georg Fischer, Jun 23 2021

A076773 2-nadirs of phi: numbers k such that phi(k-2) > phi(k-1) > phi(k) < phi(k+1) < phi(k+2).

Original entry on oeis.org

315, 525, 735, 1155, 1365, 1575, 1755, 1785, 1815, 1995, 2145, 2415, 2475, 2805, 3045, 3315, 3465, 3885, 4095, 4125, 4305, 4515, 4725, 4935, 5115, 5145, 5355, 5775, 6045, 6195, 6405, 6435, 6615, 6825, 7035, 7095, 7245, 7395, 7455, 7605, 7665, 8085

Offset: 1

Joseph L. Pe, Nov 14 2002

nonn

I call n a "k-nadir" (or nadir of depth k) of the arithmetical function f if n satisfies f(n-k) > ... > f(n-1) > f(n) < f(n+1) < ... < f(n+k).

If just phi(n-1) > phi(n) < phi(n+1) is required for odd n, does this lead to a different sequence? That is, are there consecutive odd numbers in A161962 or consecutive even numbers in A161963? - Jianing Song, Jan 12 2019

			phi(313), ..., phi(317) equal 312, 156, 144, 156, 316, respectively, so 315 is a 2-nadir of phi(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from G. C. Greubel)

Cf. A000010, A161962, A161963.

eu:=EulerPhi; f:=func; f1:= func; [k:k in [3..8100]|f(k) and f1(k)]; // Marius A. Burtea, Feb 19 2020

Select[Range[3, 10^4], EulerPhi[#-2] > EulerPhi[#-1] > EulerPhi[#] < EulerPhi[#+1] < EulerPhi[#+2] &]

[n for n in (3..9000) if euler_phi(n-2) > euler_phi(n-1) > euler_phi(n) < euler_phi(n+1) < euler_phi(n+2)] # G. C. Greubel, Feb 27 2019

A326817 Numbers k such that phi(k) > phi(k+1) > phi(k+2) > phi(k+3) where phi is the Euler totient function (A000010).

Original entry on oeis.org

823, 943, 3133, 4387, 4873, 5443, 5563, 5863, 7213, 7753, 7873, 8383, 9007, 10333, 10693, 11113, 11503, 12043, 12763, 13483, 13843, 13921, 14623, 14683, 16573, 16663, 16963, 16993, 17113, 17983, 19003, 19093, 19303, 20083, 20143, 20953, 21613, 21733, 22513

Offset: 1

Kritsada Moomuang, Oct 20 2019

nonn

			823 is in the sequence since phi(823) = 822, phi(824) = 408, phi(825) = 400, phi(826) = 348, and 822 > 408 > 400 > 348.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, page 106, entry 823.

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

Cf. A000010, A078776, A161962, A161963, A206581, A327880.

Subsequence of A328056.

aQ[n_] := AllTrue[Differences @ EulerPhi[n + Range[0, 3]], # < 0 &]; Select[Range[23000], aQ] (* Amiram Eldar, Oct 20 2019 *)

A328056 Numbers k such that phi(k) > phi(k+1) > phi(k+2) where phi is the Euler totient function (A000010).

Original entry on oeis.org

313, 523, 733, 823, 824, 943, 944, 973, 1153, 1363, 1573, 1753, 1783, 1813, 1993, 2143, 2413, 2473, 2623, 2803, 3043, 3133, 3134, 3253, 3313, 3463, 3703, 3883, 4093, 4123, 4303, 4387, 4388, 4513, 4723, 4873, 4874, 4933, 5113, 5143, 5353, 5443, 5444, 5563, 5564

Offset: 1

Kritsada Moomuang, Oct 03 2019

nonn

Contains all members k of A206581 such that k==103 (mod 210) except 103.- Robert Israel, Oct 16 2019

			313 is in the sequence since phi(313) = 312, phi(314) = 156, phi(315) = 144, and 312 > 156 > 144.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A078776, A161962, A161963, A206581, A327880.

Supersequence of A326817.

[k:k in [1..5600]| EulerPhi(k) gt EulerPhi(k+1) and EulerPhi(k+1) gt EulerPhi(k+2)]; // Marius A. Burtea, Oct 07 2019

R:= NULL: count:= 0:
q:= false: s:= 0:
for i from 1 while count < 100 do
  t:= numtheory:-phi(i);
  r:= q;
  q:= evalb(tRobert Israel, Oct 16 2019

Flatten[Position[Partition[EulerPhi[Range[5600]], 3, 1], ?(Max[Differences[#]] < 0 &)] // Quiet] (* _Amiram Eldar, Oct 06 2019 after Harvey P. Dale at A078776 *)

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A161962 Odd numbers k such that phi(k) < phi(k+1).

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A076773 2-nadirs of phi: numbers k such that phi(k-2) > phi(k-1) > phi(k) < phi(k+1) < phi(k+2).

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A326817 Numbers k such that phi(k) > phi(k+1) > phi(k+2) > phi(k+3) where phi is the Euler totient function (A000010).

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A328056 Numbers k such that phi(k) > phi(k+1) > phi(k+2) where phi is the Euler totient function (A000010).

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