A036913 Sparsely totient numbers; numbers n such that m > n implies phi(m) > phi(n).

2, 6, 12, 18, 30, 42, 60, 66, 90, 120, 126, 150, 210, 240, 270, 330, 420, 462, 510, 630, 660, 690, 840, 870, 1050, 1260, 1320, 1470, 1680, 1890, 2310, 2730, 2940, 3150, 3570, 3990, 4620, 4830, 5460, 5610, 5670, 6090, 6930, 7140, 7350, 8190, 9240, 9660

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

The paper by Masser and Shiu lists 150 terms of this sequence less than 10^6. For odd prime p, they show that p# and p*p# are in this sequence, where p# denotes the primorial (A002110). - T. D. Noe, Jun 14 2006

Conjecture: Except for 2 and 18, all terms are Zumkeller numbers (A083207). Verified for the first 1800 terms. - Ivan N. Ianakiev, Sep 04 2022

Examples

			This sequence contains 60 because of all the numbers whose totient is <=16, 60 is the largest such number. [From _Graeme McRae_, Feb 12 2009]
From _Michael De Vlieger_, Jun 25 2017: (Start)
Positions of primorials A002110(k) in a(n):
     n     k       a(n) = A002110(k)
  ----------------------------------
     1     1                       2
     2     2                       6
     5     3                      30
    13     4                     210
    31     5                    2310
    69     6                   30030
   136     7                  510510
   231     8                 9699690
   374     9               223092870
   578    10              6469693230
   836    11            200560490130
  1169    12           7420738134810
  1591    13         304250263527210
  2149    14       13082761331670030
  2831    15      614889782588491410
  3667    16    32589158477190044730
  4661    17  1922760350154212639070
(End)
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..5000
• Roger C. Baker and Glyn Harman, Sparsely totient numbers, Annales de la Faculté des Sciences de Toulouse Ser. 6, 5 no. 2 (1996), 183-190.
• Glyn Harman, On sparsely totient numbers, Glasgow Math. J. 33 (1991), 349-358.
• D. W. Masser and P. Shiu, On sparsely totient numbers, Pacific J. Math. 121, no. 2 (1986), 407-426.
• Michael De Vlieger, Largest k such that A002110(k) | a(n) and A287352(a(n)).
• Michael De Vlieger, First term m > prime(n)^2 in A036913 such that gcd(prime(n), m) = 1.

Crossrefs

Cf. A097942 (highly totient numbers). Records in A006511 (see also A132154).

Programs

Mathematica

nn=10000; lastN=Table[0,{nn}]; Do[e=EulerPhi[n]; If[e<=nn, lastN[[e]]=n], {n,10nn}]; mx=0; lst={}; Do[If[lastN[[i]]>mx, mx=lastN[[i]]; AppendTo[lst,mx]], {i,Length[lastN]}]; lst (* T. D. Noe, Jun 14 2006 *)