cp's OEIS Frontend

A286842 Least k such that the sum of proper divisors of k*n is a prime number, or -1 if no such k exists.

%I A286842 #36 Aug 06 2017 22:41:29
%S A286842 4,2,7,1,7,54,3,1,3,5,5,27,3,7,35,2,5,18,3,40,1,11,5,96,2,13,1,14,7,
%T A286842 120,5,1,99,68,1,9,3,19,1,20,5,5145000,3,88,80,23,5,48,2,1,323,52,5,6,
%U A286842 1,7,1,116,7,60,5,124,1,2,1,1650,3,34,299,35,7,32,5,37,7,19,1,26693550
%N A286842 Least k such that the sum of proper divisors of k*n is a prime number, or -1 if no such k exists.
%C A286842 Motivated by the fate of sequence A072326.
%C A286842 a(546) > 5*10^9. - _Michel Marcus_, Aug 06 2017
%C A286842 a(546) = 7975795464. When n is even the search can be sped up by observing that the exponents of the odd prime factors of n*a(n) must be even, otherwise the sum of the proper divisors n*a(n) is even and cannot be prime. So, if n is even, a(n) is equal to c*2^s*m^2, where c is the squarefree part of the odd part of n, s is 0 or 1, and m is a suitable integer. - _Giovanni Resta_, Aug 06 2017
%H A286842 Michel Marcus and Giovanni Resta, <a href="/A286842/b286842.txt">Table of n, a(n) for n = 1..1000</a> (first 545 terms from Michel Marcus)
%F A286842 a(A037020(n)) = 1.
%t A286842 Table[SelectFirst[Range[10^7], PrimeQ[DivisorSigma[1, #] - #] &[# n] &] /. k_ /; MissingQ@ k -> -1, {n, 77}] (* _Michael De Vlieger_, Aug 01 2017 *)
%o A286842 (PARI) a(n) = {my(k=1); while (!isprime(sigma(k*n)-k*n), k++); k;}
%Y A286842 Cf. A001065, A037020, A072326 (dead).
%K A286842 nonn
%O A286842 1,1
%A A286842 _Altug Alkan_, Aug 01 2017