A375262 Least positive integer m such that sigma(m)/phi(m) = n + 1/2, where sigma(.) and phi(.) are given by A000203 and A000010, respectively.

5, 459, 4, 10, 860, 18, 24, 11904, 588, 60, 1481172, 1080, 1320, 6236370, 1680, 144480, 10920, 674520, 27720, 25604040, 662535720, 1413720, 303783480, 4324320, 701205120

Offset: 1

Author

Zhi-Wei Sun, Aug 08 2024

Keywords

• nonn
• more

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

Examples

			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.
		

Links

• B.S.K.R. Somayajulu, The sequence sigma(n)/phi(n), Math. Student, Vol. 45, No. 1 (1977), pp. 52-54; entire issue.
• 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.

Crossrefs

Cf. A000010, A000203, A055234, A065824.

Programs

Mathematica

sigma[n_]:=sigma[n]=DivisorSigma[1,n]; phi[n_]:=phi[n]=EulerPhi[n];
tab={};Do[m=1;While[sigma[m]/phi[m]!=n+1/2,m=m+1];tab=Append[tab,m],{n,1,20}];Print[tab]

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

Extensions

a(21)-a(24) from Amiram Eldar, Aug 08 2024

a(25) from Chai Wah Wu, Aug 12 2024