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A065824 Smallest solution m to (n+1)*phi(m) = n*sigma(m), or -1 if no solution exists.

%I A065824 #28 Aug 12 2024 12:02:32
%S A065824 3,5,7,323,11,13,899,17,19,1763,23,5249,3239,29,31,979801,5459,37,
%T A065824 10763,41,43,9179,47,9701,10403,53,12319,5646547,59,61,24569,19109,67,
%U A065824 19043,71,73,22499,50819,79,41309,83,32639,46979,89,34579,39059,125969
%N A065824 Smallest solution m to (n+1)*phi(m) = n*sigma(m), or -1 if no solution exists.
%C A065824 If p = a(n) is a prime solution, then (n+1)*(p-1) = n*(p+1) and p = 2n+1, so position for p if it is in fact a minimal solution is at n = (p-1)/2. E.g. 29 appears at 14th position shown by A005097. On the other hand large and (seemingly always composite) solutions arise at indices shown essentially by A047845. Also, differences between the sites of two consecutive small prime solutions appears to be d/2, half the difference between consecutive primes (A001223).
%H A065824 Donovan Johnson, <a href="/A065824/b065824.txt">Table of n, a(n) for n = 1..456</a>
%F A065824 (n+1)*A000010(a(n)) = n*A000203(a(n)), smallest x=a(n) solutions.
%t A065824 max = 10^7; a[n_] := For[m = 3, m <= max, m++, If[(n+1)*EulerPhi[m] == n*DivisorSigma[1, m], Print[m]; Return[m]]] /. Null -> -1; Array[a, 50] (* _Jean-François Alcover_, Oct 08 2016 *)
%o A065824 (Python)
%o A065824 from itertools import count
%o A065824 from math import prod
%o A065824 from sympy import factorint
%o A065824 def A065824(n):
%o A065824     for m in count(1):
%o A065824         f = factorint(m)
%o A065824         if (n+1)*m*prod((p-1)**2 for p in f)==n*prod(p**(e+2)-p for p,e in f.items()):
%o A065824             return m # _Chai Wah Wu_, Aug 12 2024
%Y A065824 Cf. A000010, A000203, A062699, A065818, A065819, A065822, A065823.
%Y A065824 See also A005097, A047845, A014076, A001223.
%K A065824 nice,nonn
%O A065824 1,1
%A A065824 _Labos Elemer_, Nov 23 2001