A135718 -id:A135718 - cp's OEIS frontend

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

Offset: 2

Franklin T. Adams-Watters, Aug 27 2011

nonn

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

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

Cf. A000203, A007978, A027750, A135718.

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

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

a193574[n_] := First[Select[Divisors[DivisorSigma[1, n]], Mod[n, #]!=0&]]
Map[a193574, Range[2, 80]] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)

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

A160995 The smallest positive integer neither a divisor of n nor coprime to n.

Original entry on oeis.org

4, 6, 6, 10, 4, 14, 6, 6, 4, 22, 8, 26, 4, 6, 6, 34, 4, 38, 6, 6, 4, 46, 9, 10, 4, 6, 6, 58, 4, 62, 6, 6, 4, 10, 8, 74, 4, 6, 6, 82, 4, 86, 6, 6, 4, 94, 9, 14, 4, 6, 6, 106, 4, 10, 6, 6, 4, 118, 8, 122, 4, 6, 6, 10, 4, 134, 6, 6, 4, 142, 10, 146, 4, 6, 6, 14, 4, 158, 6, 6, 4, 166, 8, 10, 4, 6

Offset: 2

Leroy Quet, Jun 01 2009

nonn

a(1) doesn't exist because 1 is coprime to all integers.

Terms are composite since primes either divide or are coprime to other numbers. - Michael De Vlieger, Feb 20 2025

			From _David James Sycamore_, Feb 28 2025: (Start)
Using my formula above: n = 4235 = 5*7*11^2, so a(n) = 2*5 = 10.
For n = odd prime p, a(n) = 2*p.
For n = 2, a(n) = min{2^2, 2*3} = 4.
For n = 4, a(n) = min{2^3, 2*3} = 6. (For all n = 2^k, k >= 2, a(n) = 6.)
For n = 120 = 2^3*3*5, a(n) = min{16, 9, 25, 14} = 9.
For n = 5040 = 2^4*3^2*5*7, a(n) = min{32, 27, 25, 49, 22} = 22.
For n = 3603600 = 2^4*3^2*5^2*7*11*13, a(n) = min{32,27,125,49,121,169,34} = 27. (End)

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000

Cf. A001221, A020639, A053669, A096014, A100484, A133995, A135718.

Table[k = 3; Until[1 < GCD[k, n] < k, k++]; k, {n, 2, 120}] (* Michael De Vlieger, Feb 20 2025 *)

a(n)=for(k=4,2*n,if(gcd(n,k)>1 && n%k, return(k))) \\ Charles R Greathouse IV, Apr 05 2013

a(n)=my(f=factor(n),b);forprime(p=2,,if(n%p,b=p*f[1,1];break));for(i=1,#f[,1],b=min(b,f[i,1]^(f[i,2]+1)));b \\ Charles R Greathouse IV, Apr 05 2013

For composite n > 4, a(n) is the first term of row n of A133995. - Michael De Vlieger, Feb 20 2025
For even n whose prime factorization is Product_{i=1..k} (p_i)^(e_i), a(n) = min({p_i^(e_i + 1) : i = 1..k} U {2*q}), where q = A053669(n); for odd n, a(n) = 2*A020639(n); see Example. - David James Sycamore, Feb 28 2025 [edited by Peter Munn, Jul 20 2025]
a(n) = min(A096014(n), A135718(n)). - Michael De Vlieger, Feb 24 2025

Extended by Ray Chandler, Jun 13 2009

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

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A160995 The smallest positive integer neither a divisor of n nor coprime to n.

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