OFFSET
1,1
COMMENTS
Conjecture: Any rational number r >= 1 can be written as sigma(m)/phi(m) with m a positive integer.
We have verified this for rational numbers a/b with 36 >= a >= b >= 1.
In 1977, B.S.K.R. Somayajulu proved that the set {sigma(n)/phi(n): n = 1,2,3,...} is dense in the interval (1,+oo).
a(27) = 790269480. - Chai Wah Wu, Aug 12 2024
REFERENCES
B.S.K.R. Somayajulu, The sequence sigma(n)/phi(n), Math. Student 45 (1977), 52-54.
LINKS
Zhi-Wei Sun Is it true that {sigma(n)/phi(n): n >= 1} = {r in Q: r >= 1}? Question 476578 at MathOverflow, August 8, 2024.
EXAMPLE
a(1) = 5 with sigma(5)/phi(5) = 6/4 = 1 + 1/2.
a(2) = 459 = 3^3*17 with sigma(459)/phi(459) = 720/288 = 2 + 1/2.
a(20) = 25604040 = 2^3*3*5*7*11*17*163 with sigma(25604040)/phi(25604040) = 102021120/4976640 = 20 + 1/2.
MATHEMATICA
sigma[n_]:=sigma[n]=DivisorSigma[1, n]; phi[n_]:=phi[n]=EulerPhi[n];
tab=}; Do[m=1; Label[aa]; If[sigma[m]/phi[m]==n+1/2, tab=Append[tab, m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb], {n, 1, 20}]; Print[tab]
PROG
(PARI) a(n) = my(k=1); while (sigma(k)/eulerphi(k) != n + 1/2, k++); k; \\ Michel Marcus, Aug 08 2024
(Python)
from itertools import count
from math import prod
from sympy import factorint
def A375262(n):
for m in count(1):
f = factorint(m)
if ((n<<1)+1)*m*prod((p-1)**2 for p in f)==prod(p**(e+2)-p for p, e in f.items())<<1:
return m # Chai Wah Wu, Aug 11 2024
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Zhi-Wei Sun, Aug 08 2024
EXTENSIONS
a(21)-a(24) from Amiram Eldar, Aug 08 2024
a(25) from Chai Wah Wu, Aug 12 2024
STATUS
approved