A018894 Numbers k such that sigma(k)/phi(k) 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

Author

Michel ten Voorde

Keywords

• nonn

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(2*m) = phi(m) and sigma(2*m) = 3*sigma(m).

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

So sigma(2*m)/phi(2*m) > 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

Crossrefs

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

Programs

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 *)

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

Extensions

More terms from Jud McCranie, Nov 09 2001

Initial term added by Arkadiusz Wesolowski, Sep 06 2012