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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.

%I A375262 #39 Jun 21 2025 07:54:18
%S A375262 5,459,4,10,860,18,24,11904,588,60,1481172,1080,1320,6236370,1680,
%T A375262 144480,10920,674520,27720,25604040,662535720,1413720,303783480,
%U A375262 4324320,701205120
%N 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.
%C A375262 Conjecture: Any rational number r >= 1 can be written as sigma(m)/phi(m) with m a positive integer.
%C A375262 We have verified this for rational numbers a/b with 36 >= a >= b >= 1.
%C A375262 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 A375262 a(27) = 790269480. - _Chai Wah Wu_, Aug 12 2024
%H A375262 B.S.K.R. Somayajulu, The sequence sigma(n)/phi(n), Math. Student, Vol. 45, No. 1 (1977), pp. 52-54; <a href="https://schoolbooksarchive.azimpremjiuniversity.edu.in/handle/20.500.12497/11731">entire issue</a>.
%H A375262 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 A375262 a(1) = 5 with sigma(5)/phi(5) = 6/4 = 1 + 1/2.
%e A375262 a(2) = 459 = 3^3*17 with sigma(459)/phi(459) = 720/288 = 2 + 1/2.
%e A375262 a(20) = 25604040 = 2^3*3*5*7*11*17*163 with sigma(25604040)/phi(25604040) = 102021120/4976640 = 20 + 1/2.
%t A375262 sigma[n_]:=sigma[n]=DivisorSigma[1,n]; phi[n_]:=phi[n]=EulerPhi[n];
%t A375262 tab={};Do[m=1;While[sigma[m]/phi[m]!=n+1/2,m=m+1];tab=Append[tab,m],{n,1,20}];Print[tab]
%o A375262 (PARI) a(n) = my(k=1); while (sigma(k)/eulerphi(k) != n + 1/2, k++); k; \\ _Michel Marcus_, Aug 08 2024
%o A375262 (Python)
%o A375262 from itertools import count
%o A375262 from math import prod
%o A375262 from sympy import factorint
%o A375262 def A375262(n):
%o A375262     for m in count(1):
%o A375262         f = factorint(m)
%o A375262         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 A375262             return m # _Chai Wah Wu_, Aug 11 2024
%Y A375262 Cf. A000010, A000203, A055234, A065824.
%K A375262 nonn,more
%O A375262 1,1
%A A375262 _Zhi-Wei Sun_, Aug 08 2024
%E A375262 a(21)-a(24) from _Amiram Eldar_, Aug 08 2024
%E A375262 a(25) from _Chai Wah Wu_, Aug 12 2024