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A018894 Numbers k such that sigma(k)/phi(k) sets a new record.

%I A018894 #88 Aug 14 2025 06:02:45
%S A018894 1,2,4,6,12,24,30,60,120,180,210,360,420,840,1260,1680,2520,4620,9240,
%T A018894 13860,18480,27720,55440,110880,120120,180180,240240,360360,720720,
%U A018894 1441440,2162160,3603600,4084080,4324320,6126120,12252240,24504480,36756720,61261200
%N A018894 Numbers k such that sigma(k)/phi(k) sets a new record.
%C A018894 Remarkably similar to but ultimately different from A126098. - Jorg Brown and _N. J. A. Sloane_, Mar 06 2007
%C A018894 Is a(n+1) <= 2*a(n)? Is a(n) divisible by the primorial p# where p is the largest prime divisor of a(n)? Is a(k) divisible by p# for all k > n + 1? (Cf. A002110.) - _David A. Corneth_, May 22 2016
%C A018894 From _Jud McCranie_, Nov 28 2017: (Start)
%C A018894 Yes, a(n+1) <= 2*a(n) -- if m is odd, phi(2*m) = phi(m) and sigma(2*m) = 3*sigma(m).
%C A018894 If m is even then phi(2*m) = 2*phi(m) and sigma(2*m) > 2*sigma(m).
%C A018894 So sigma(2*m)/phi(2*m) > sigma(m)/phi(m). (End)
%C A018894 From _David A. Corneth_, Sep 10 2020: (Start)
%C A018894 Subsequence of A025487.
%C A018894 Let prime(n)# be the product of the first n primes. Then the LCM of the terms <= 10^40 is 89# * 7# * 5# * (3#)^2 * (2#)^4.
%C A018894 We can assume a larger LCM for terms <= 10^60 namely P# * (13#)^3 * (11#) * (5#) * (3#)^2 * (2#)^4. This gives a total of 466 terms <= 10^75 where P is an arbitrary large prime such that P# <= 10^75.
%C A018894 The LCM of these found terms is a proper divisor and for all primes p <= 13 the exponent is less than the assumed prime. Conjecture: These 466 terms are the terms <= 10^75.
%C A018894 For all 240 terms 1 < t <= 10^40 the following holds: there exists a p|t such that t/p is a term. Conjecture: This holds for all terms t > 1.
%C A018894 Using this technique to find terms I get 6522 terms <= 10^1000 and no conflict with terms found above.
%C A018894 See attached file with terms assuming these conjectures. (End)
%H A018894 David A. Corneth, <a href="/A018894/b018894.txt">Table of n, a(n) for n = 1..241</a> (first 79 terms from Jud McCranie)
%H A018894 Jorg Brown, <a href="/A018894/a018894.txt">Comparison of records in sigma(n)/phi(n) and A018892</a>
%H A018894 David A. Corneth, <a href="/A018894/a018894.gp.txt">Conjectured 6522 terms <= 10^1000</a>
%t A018894 Flatten@ Function[k, FirstPosition[k, #] & /@ Union@ Rest@ FoldList[Max, 0, k]]@ Array[DivisorSigma[1, #]/EulerPhi@ # &, 10^7] (* _Michael De Vlieger_, May 27 2016, Version 10 *)
%o A018894 (PARI) lista(nn) = {mse = 0; for (n=1, nn, se = sigma(n)/eulerphi(n); if (se > mse, print1(n, ", "); mse = se););} \\ _Michel Marcus_, Jul 10 2015
%Y A018894 Cf. A000010, A000203, A015702, A020492, A025487, A126098, A002110.
%K A018894 nonn
%O A018894 1,2
%A A018894 _Michel ten Voorde_
%E A018894 More terms from _Jud McCranie_, Nov 09 2001
%E A018894 Initial term added by _Arkadiusz Wesolowski_, Sep 06 2012