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

1, 2, 4, 6, 12, 24, 30, 60, 120, 180, 210, 360, 420, 840, 1260, 1680, 2520, 4620, 9240, 13860, 18480, 27720, 55440, 110880, 120120, 180180, 240240, 360360, 720720, 1441440, 2162160, 3603600, 4084080, 4324320, 6126120, 12252240, 24504480, 36756720, 61261200

Offset

1,2

Comments

Remarkably similar to but ultimately different from A126098. - Jorg Brown and N. J. A. Sloane, Mar 06 2007

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

From Jud McCranie, Nov 28 2017: (Start)

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

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

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

From David A. Corneth, Sep 10 2020: (Start)

Subsequence of A025487.

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.

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.

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.

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.

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

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

Links

David A. Corneth, Table of n, a(n) for n = 1..241 (first 79 terms from Jud McCranie)

Jorg Brown, Comparison of records in sigma(n)/phi(n) and A018892

David A. Corneth, Conjectured 6522 terms <= 10^1000

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000010, A000203, A015702, A020492, A025487, A126098, A002110.

Sequence in context: A362081 A265719 A126098 * A340816 A168264 A282472

Adjacent sequences: A018891 A018892 A018893 * A018895 A018896 A018897

Keyword

nonn

Author

Michel ten Voorde

Extensions

More terms from Jud McCranie, Nov 09 2001

Initial term added by Arkadiusz Wesolowski, Sep 06 2012

Status

approved