A161962 -id:A161962 - 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

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

A327880 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

1484, 2534, 3002, 3674, 3926, 4454, 4484, 4784, 4844, 5264, 5312, 5984, 6104, 7994, 8294, 8414, 8774, 8834, 9074, 9164, 9944, 10004, 10724, 11024, 11684, 11894, 12254, 13034, 13064, 13166, 13454, 13754, 14234, 15344, 15554, 16184, 16214, 16814, 17384, 17534

Offset: 1

Amiram Eldar, Sep 28 2019

nonn

			1484 is in the sequence since phi(1484) = 624, phi(1485) = 720, phi(1486) = 742, and phi(1487) = 1486, and 624 < 720 < 742 < 1486.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 34, entry 105 and p. 130, entry 1484.

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

Cf. A000010, A078776, A161962.

aQ[n_] := AllTrue[Differences @ EulerPhi[n + Range[0, 3]], # > 0 &]; Select[Range[18000], aQ]

ok(k)={for(i=0, 2, if(eulerphi(k+i) >= eulerphi(k+i+1), return(0))); 1}
{ select(ok, [1..20000]) } \\ Andrew Howroyd, Sep 28 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 *)

A076771 Even numbers n representable as the sum of two non-coprime odd composites.

Original entry on oeis.org

18, 24, 30, 36, 40, 42, 48, 50, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 108, 110, 112, 114, 120, 126, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174, 176, 180, 182, 184, 186, 190, 192, 196, 198, 200, 204

Offset: 1

Jon Perry, Nov 14 2002

It is conjectured that there is no N such that for all n > N, every even number is represented.
The conjecture is true since 2p, p prime, is never a member of this sequence. - Charles R Greathouse IV, Aug 08 2011
An equivalent definition of this sequence: even numbers n such that phi(n) < (n-4)/2. - Arkadiusz Wesolowski, Aug 08 2011
Also: products of an even number >= 6 and an odd number >= 3. - Charles R Greathouse IV, Aug 08 2011

			40 = 5*(3 + 5).

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A161962.

upTo(lim)=my(v=List());forstep(a=6,lim\3,2,forstep(b=3,lim\a,2,listput(v,a*b)));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Aug 08 2011

upTo(lim)=my(v=List(),p=7,m);forprime(q=8,lim\2,forstep(n=p+2,q-2,2,m=n;while((m<<=1)<=lim,listput(v,m)));p=q);forstep(n=2*p+4,lim,4,listput(v,n));forprime(p=3,lim>>3,m=p<<3;while(m<=lim,listput(v,m);m<<=1));vecsort(Vec(v)) \\ Charles R Greathouse IV, Aug 08 2011

Offset corrected by Arkadiusz Wesolowski, Aug 08 2011

A179271 Odd long legs `B` of more than one primitive Pythagorean triangle.

Original entry on oeis.org

2145, 3315, 3465, 4095, 4845, 5005, 5865, 6435, 6545, 6555, 7735, 8645, 9009, 9945, 10005, 10695, 11305, 11781, 13167, 13485, 13685, 13923, 14535, 15015, 15295, 15561, 16065, 16095, 17017, 17205, 17255, 17835, 17955, 18837, 19019, 19065

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 06 2010

nonn

2145,752,2273;2145,1568,2657;;3315,812,3413;3315,2852,4373;

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A024407, A024408, A024409, A024410, A024411, A161962, A174830

lst1={};lst={0};Do[Do[If[GCD[a,b]==1,c=Sqrt[a^2+b^2];If[IntegerQ[c],AppendTo[lst,b];L=Length[lst];If[lst[[L]]==lst[[L-1]]&&OddQ[lst[[L]]],Print[lst[[L]]];AppendTo[lst1,lst[[L]]]]]],{a,b-1,3,-1}],{b,4,4*7!}];lst1

a(20) - a(36) Robert G. Wilson v, Jul 12 2010

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A161963 Even numbers n for which phi(n) > phi(n+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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A327880 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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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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A076771 Even numbers n representable as the sum of two non-coprime odd composites.

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A179271 Odd long legs `B` of more than one primitive Pythagorean triangle.

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