A018894 - OEIS

%I #85 Sep 11 2020 10:57:53

%S 1,2,4,6,12,24,30,60,120,180,210,360,420,840,1260,1680,2520,4620,9240,

%T 13860,18480,27720,55440,110880,120120,180180,240240,360360,720720,

%U 1441440,2162160,3603600,4084080,4324320,6126120,12252240,24504480,36756720,61261200

%N Numbers n such that sigma(n)/phi(n) sets a new record.

%C Remarkably similar to but ultimately different from A126098. - Jorg Brown and _N. J. A. Sloane_, Mar 06 2007

%C 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 From _Jud McCranie_, Nov 28 2017: (Start)

%C Yes, a(n+1) <= 2*a(n) -- if m is odd, phi(2m) = phi(m) and sigma(2m) = 3*sigma(m).

%C If m is even then phi(2m) = 2*phi(m) and sigma(2m) > 2*sigma(m).

%C So sigma(2m)/phi(2m) > sigma(m)/phi(m). (end)

%C From _David A. Corneth_, Sep 10 2020: (Start)

%C Subsequence of A025487.

%C 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 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 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 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 Using this technique to find terms I get 6522 terms <= 10^1000 and no conflict with terms found above.

%C See attached file with terms assuming these conjectures. (End)

%H 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 Jorg Brown, <a href="/A018894/a018894.txt">Comparison of records in sigma(n)/phi(n) and A018892</a>

%H David A. Corneth, <a href="/A018894/a018894.gp.txt">Conjectured 6522 terms <= 10^1000</a>

%t 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 (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 Cf. A000010, A000203, A015702, A020492, A025487, A126098, A002110.

%K nonn

%O 1,2

%A _Michel ten Voorde_

%E More terms from _Jud McCranie_, Nov 09 2001

%E Initial term added by _Arkadiusz Wesolowski_, Sep 06 2012