A193574 - OEIS

A193574

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

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, 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, 2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 4, 2, 3

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OFFSET

2,1

COMMENTS

a(n) = 2 iff n is an odd number that is not a perfect square.

From Hartmut F. W. Hoft, May 05 2017: (Start)

(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.

(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.

(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.

(4) Conjecture: a(n) > n implies a(n) = sigma(n); tested through n = 20000000.

(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)

(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.

(End)

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

MATHEMATICA

a193574[n_] := First[Select[Divisors[DivisorSigma[1, n]], Mod[n, #]!=0&]]

Map[a193574, Range[2, 80]] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)

PROG

(PARI) a(n)=local(ds); ds=divisors(sigma(n)); for(k=2, #ds, if(n%ds[k], return(ds[k])))

(Haskell)

import Data.List ((\\))

a193574 n = head [d | d <- [1..sigma] \\ nDivisors, mod sigma d == 0]

where nDivisors = a027750_row n

sigma = sum nDivisors