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%I #43 Apr 13 2023 09:25:26
%S 2,6,12,18,30,42,60,66,90,120,126,150,210,240,270,330,420,462,510,630,
%T 660,690,840,870,1050,1260,1320,1470,1680,1890,2310,2730,2940,3150,
%U 3570,3990,4620,4830,5460,5610,5670,6090,6930,7140,7350,8190,9240,9660
%N Sparsely totient numbers; numbers n such that m > n implies phi(m) > phi(n).
%C 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
%C Conjecture: Except for 2 and 18, all terms are Zumkeller numbers (A083207). Verified for the first 1800 terms. - _Ivan N. Ianakiev_, Sep 04 2022
%H T. D. Noe, <a href="/A036913/b036913.txt">Table of n, a(n) for n = 1..5000</a>
%H Roger C. Baker and Glyn Harman, <a href="http://www.numdam.org/item?id=AFST_1996_6_5_2_183_0">Sparsely totient numbers</a>, Annales de la Faculté des Sciences de Toulouse Ser. 6, 5 no. 2 (1996), 183-190.
%H Glyn Harman, <a href="https://doi.org/10.1017/S0017089500008417">On sparsely totient numbers</a>, Glasgow Math. J. 33 (1991), 349-358.
%H D. W. Masser and P. Shiu, <a href="https://projecteuclid.org/download/pdf_1/euclid.pjm/1102702441">On sparsely totient numbers</a>, Pacific J. Math. 121, no. 2 (1986), 407-426.
%H Michael De Vlieger, <a href="/A036913/a036913.txt">Largest k such that A002110(k) | a(n) and A287352(a(n))</a>.
%H Michael De Vlieger, <a href="/A036913/a036913_1.txt">First term m > prime(n)^2 in A036913 such that gcd(prime(n), m) = 1</a>.
%e 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]
%e From _Michael De Vlieger_, Jun 25 2017: (Start)
%e Positions of primorials A002110(k) in a(n):
%e n k a(n) = A002110(k)
%e ----------------------------------
%e 1 1 2
%e 2 2 6
%e 5 3 30
%e 13 4 210
%e 31 5 2310
%e 69 6 30030
%e 136 7 510510
%e 231 8 9699690
%e 374 9 223092870
%e 578 10 6469693230
%e 836 11 200560490130
%e 1169 12 7420738134810
%e 1591 13 304250263527210
%e 2149 14 13082761331670030
%e 2831 15 614889782588491410
%e 3667 16 32589158477190044730
%e 4661 17 1922760350154212639070
%e (End)
%t 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 *)
%Y Cf. A097942 (highly totient numbers). Records in A006511 (see also A132154).
%K nonn
%O 1,1
%A _David W. Wilson_