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A193574 Smallest divisor of sigma(n) that does not divide n.

%I A193574 #35 May 12 2017 01:01:34
%S A193574 3,2,7,2,4,2,3,13,3,2,7,2,3,2,31,2,13,2,3,2,3,2,5,31,3,2,8,2,4,2,3,2,
%T A193574 3,2,7,2,3,2,3,2,4,2,3,2,3,2,31,3,3,2,7,2,4,2,3,2,3,2,7,2,3,2,127,2,4,
%U A193574 2,3,2,3,2,5,2,3,2,5,2,4,2,3
%N A193574 Smallest divisor of sigma(n) that does not divide n.
%C A193574 a(n) = 2 iff n is an odd number that is not a perfect square.
%C A193574 From _Hartmut F. W. Hoft_, May 05 2017: (Start)
%C A193574 (1)  Every a(n) > n is a prime: Because of the minimality of a(n), a(n) = u*v with gcd(u,v)=1 leads to the contradiction (u*v)|n. Similarly, a(n)=p^k with p prime an k>1 leads to the contradiction (p^k-1)/(p-1) | n.
%C A193574 (2)  n=p^(2*k), k>=1 and 2*k+1 prime, when a(n) = sigma(n) for n>2: Because n having two distinct prime factors implies sigma(n) composite, and if n is an odd power of a prime then 2|sigma(n). Finally, if 2*k+1=u*v  with  u,v > 1 then sigma(p^(u-1)) divides sigma(p^(2*k)), but not p^(2k), for any prime p, contradicting minimality of a(n). For example, no number sigma(p^8) for any prime p is in the sequence.
%C A193574 (3)  The converse of (2) is false since, e.g. sigma(7^2) = 3*19 so that a(7^2) = 3, and sigma(2^10) = 23*89 so that a(2^10) = 23.
%C A193574 (4)  Conjecture: a(n) > n implies a(n) = sigma(n); tested through n = 20000000.
%C A193574 (5)  Subsequences are: A053183 (sigma(p^2) is prime for prime p), A190527 (sigma(p^4) is prime for prime p), A194257 (sigma(p^6) is prime for prime p), A286301 (sigma(p^10) is prime for prime p)
%C A193574 (6)  Subsequences are: A000668 (primes of form 2^p-1), A076481 (primes of form (3^p-1)/2), A086122 (primes of form (5^p-1)/4), A102170 (primes of form (7^p-1)/6), all when p is prime.
%C A193574 (End)
%C A193574 Up to n = 10^6, there are 89 distinct elements. For those n, a(n) is prime. If it's not, it's a power of 2, a power of 3 or a perfect square <= 121. - _David A. Corneth_, May 10 2017
%H A193574 Reinhard Zumkeller, <a href="/A193574/b193574.txt">Table of n, a(n) for n = 2..10000</a>
%t A193574 a193574[n_] := First[Select[Divisors[DivisorSigma[1, n]], Mod[n, #]!=0&]]
%t A193574 Map[a193574, Range[2, 80]] (* data *) (* _Hartmut F. W. Hoft_, May 05 2017 *)
%o A193574 (PARI) a(n)=local(ds);ds=divisors(sigma(n));for(k=2,#ds,if(n%ds[k],return(ds[k])))
%o A193574 (Haskell)
%o A193574 import Data.List ((\\))
%o A193574 a193574 n = head [d | d <- [1..sigma] \\ nDivisors, mod sigma d == 0]
%o A193574    where nDivisors = a027750_row n
%o A193574          sigma = sum nDivisors
%o A193574 -- _Reinhard Zumkeller_, May 20 2015, Aug 28 2011
%Y A193574 Cf. A000203, A007978, A027750, A135718.
%Y A193574 Subsequences: A000668, A053183, A076481, A086122, A102170, A190527, A194257, A286301.
%K A193574 nonn
%O A193574 2,1
%A A193574 _Franklin T. Adams-Watters_, Aug 27 2011