%I #50 Aug 12 2024 13:06:04
%S 1,3,2,14,56,6,12,42,30,168,2580,210,630,420,840,20790,416640,9240,
%T 291060,83160,120120,5165160,1719277560,43825320,26860680,277560360,
%U 1304863560,569729160,587133466920,16522145640,33044291280,563462139240,1140028049160,9015394227840,1255683068640,65361608151840
%N Smallest x such that sigma(x) = n*phi(x), or -1 if no such x exists.
%C Conjecture: For each n, a(n) > 0. - _Farideh Firoozbakht_, Sep 12 2004
%C a(33) > 10^12. - _Donovan Johnson_, Mar 06 2012
%C a(34) <= 9015394227840, a(35) <= 1255683068640. - _Giovanni Resta_, May 08 2017
%C 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
%C a(38) = 299761858075680, a(39) = 413956851628320. a(37), a(40), and higher terms are > 4.2*10^14. - _Jud McCranie_, Nov 27 2017
%C a(37), a(40), and higher terms are > 6.0 x 10^14. - _Jud McCranie_, Dec 27 2017
%F a(n) = Min{x : A000203(x)/A000010(x) = n} = Min{x : A023897(x) = n}
%e sigma(14) = 24 = 4*phi(14), so a(4) = 14.
%e n = 21: a(21) = 120120 = 2*2*2*3*5*7*11*13, sigma(120120) = 483840 = n*phi(120120), phi(120120) = 23040.
%t a[n_]:=(For[m=1,DivisorSigma[1,m]!=n EulerPhi[m],m++ ];m);Do[Print[a[n]], {n,31}] (* _Farideh Firoozbakht_, Oct 31 2008 *)
%o (PARI) a(n) = {k = 1; while(sigma(k) != n*eulerphi(k), k++); k;} \\ _Michel Marcus_, Sep 01 2014
%o (Python)
%o from math import prod
%o from itertools import count
%o from sympy import factorint
%o def A055234(n):
%o for m in count(1):
%o f = factorint(m)
%o if n*m*prod((p-1)**2 for p in f)==prod(p**(e+2)-p for p,e in f.items()):
%o return m # _Chai Wah Wu_, Aug 12 2024
%Y Cf. A000010, A000203, A020492, A023897, A088830.
%K nonn
%O 1,2
%A _Jud McCranie_, Jun 21 2000
%E More terms from _Farideh Firoozbakht_, Sep 12 2004
%E a(32) from _Donovan Johnson_, Mar 06 2012
%E a(33) from _Giovanni Resta_, May 08 2017
%E a(34)-a(36) from _Jud McCranie_, Sep 10 2017