A375262 - OEIS

%I #31 Aug 12 2024 12:03:03

%S 5,459,4,10,860,18,24,11904,588,60,1481172,1080,1320,6236370,1680,

%T 144480,10920,674520,27720,25604040,662535720,1413720,303783480,

%U 4324320,701205120

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

%C Conjecture: Any rational number r >= 1 can be written as sigma(m)/phi(m) with m a positive integer.

%C We have verified this for rational numbers a/b with 36 >= a >= b >= 1.

%C 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).

%C a(27) = 790269480. - _Chai Wah Wu_, Aug 12 2024

%D B.S.K.R. Somayajulu, The sequence sigma(n)/phi(n), Math. Student 45 (1977), 52-54.

%H Zhi-Wei Sun <a href="https://mathoverflow.net/questions/476578">Is it true that {sigma(n)/phi(n): n >= 1} = {r in Q: r >= 1}?</a> Question 476578 at MathOverflow, August 8, 2024.

%e a(1) = 5 with sigma(5)/phi(5) = 6/4 = 1 + 1/2.

%e a(2) = 459 = 3^3*17 with sigma(459)/phi(459) = 720/288 = 2 + 1/2.

%e a(20) = 25604040 = 2^3*3*5*7*11*17*163 with sigma(25604040)/phi(25604040) = 102021120/4976640 = 20 + 1/2.

%t sigma[n_]:=sigma[n]=DivisorSigma[1,n]; phi[n_]:=phi[n]=EulerPhi[n];

%t 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]

%o (PARI) a(n) = my(k=1); while (sigma(k)/eulerphi(k) != n + 1/2, k++); k; \\ _Michel Marcus_, Aug 08 2024

%o (Python)

%o from itertools import count

%o from math import prod

%o from sympy import factorint

%o def A375262(n):

%o     for m in count(1):

%o         f = factorint(m)

%o         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:

%o             return m # _Chai Wah Wu_, Aug 11 2024

%Y Cf. A000010, A000203, A055234, A065824.

%K nonn,more

%O 1,1

%A _Zhi-Wei Sun_, Aug 08 2024

%E a(21)-a(24) from _Amiram Eldar_, Aug 08 2024

%E a(25) from _Chai Wah Wu_, Aug 12 2024