A055234 Smallest x such that sigma(x) = n*phi(x), or -1 if no such x exists.

1, 3, 2, 14, 56, 6, 12, 42, 30, 168, 2580, 210, 630, 420, 840, 20790, 416640, 9240, 291060, 83160, 120120, 5165160, 1719277560, 43825320, 26860680, 277560360, 1304863560, 569729160, 587133466920, 16522145640, 33044291280, 563462139240, 1140028049160, 9015394227840, 1255683068640, 65361608151840

Offset: 1

Author

Jud McCranie, Jun 21 2000

Keywords

• nonn

Comments

Conjecture: For each n, a(n) > 0. - Farideh Firoozbakht, Sep 12 2004

a(33) > 10^12. - Donovan Johnson, Mar 06 2012

a(34) <= 9015394227840, a(35) <= 1255683068640. - Giovanni Resta, May 08 2017

Terms after a(36) are > 10^14. a(37) <= 4771397395084320, a(38) <= 2418379501618080, a(39) <= 413956851628320, a(40) <= 1241870554884960, and a(42) <= 50916692750283360. - Jud McCranie, Sep 13 2017

a(38) = 299761858075680, a(39) = 413956851628320. a(37), a(40), and higher terms are > 4.2*10^14. - Jud McCranie, Nov 27 2017

a(37), a(40), and higher terms are > 6.0 x 10^14. - Jud McCranie, Dec 27 2017

Examples

			sigma(14) = 24 = 4*phi(14), so a(4) = 14.
n = 21: a(21) = 120120 = 2*2*2*3*5*7*11*13, sigma(120120) = 483840 = n*phi(120120), phi(120120) = 23040.
		

Crossrefs

Cf. A000010, A000203, A020492, A023897, A088830.

Programs

Mathematica

a[n_]:=(For[m=1,DivisorSigma[1,m]!=n EulerPhi[m],m++ ];m);Do[Print[a[n]], {n,31}] (* Farideh Firoozbakht, Oct 31 2008 *)

PARI

a(n) = {k = 1; while(sigma(k) != n*eulerphi(k), k++); k;} \\ Michel Marcus, Sep 01 2014

Python

from math import prod
from itertools import count
from sympy import factorint
def A055234(n):
    for m in count(1):
        f = factorint(m)
        if n*m*prod((p-1)**2 for p in f)==prod(p**(e+2)-p for p,e in f.items()):
            return m # Chai Wah Wu, Aug 12 2024

Formula

a(n) = Min{x : A000203(x)/A000010(x) = n} = Min{x : A023897(x) = n}

Extensions

More terms from Farideh Firoozbakht, Sep 12 2004

a(32) from Donovan Johnson, Mar 06 2012

a(33) from Giovanni Resta, May 08 2017

a(34)-a(36) from Jud McCranie, Sep 10 2017