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A193574 Smallest divisor of sigma(n) that does not divide n. 5
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 (list; graph; refs; listen; history; text; internal format)
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

-- Reinhard Zumkeller, May 20 2015, Aug 28 2011

CROSSREFS

Cf. A000203, A007978, A027750, A135718.

Subsequences: A000668, A053183, A076481, A086122, A102170, A190527, A194257, A286301.

Sequence in context: A137822 A300845 A302714 * A209639 A174238 A175920

Adjacent sequences:  A193571 A193572 A193573 * A193575 A193576 A193577

KEYWORD

nonn

AUTHOR

Franklin T. Adams-Watters, Aug 27 2011

STATUS

approved

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Last modified June 25 12:55 EDT 2019. Contains 324352 sequences. (Running on oeis4.)