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

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, 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, 1, 7, 1, 116, 7, 60, 5, 124, 1, 2, 1, 1650, 3, 34, 299, 35, 7, 32, 5, 37, 7, 19, 1, 26693550

Offset: 1

Author

Altug Alkan, Aug 01 2017

Keywords

• nonn

Comments

Motivated by the fate of sequence A072326.

a(546) > 5*10^9. - Michel Marcus, Aug 06 2017

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

Links

• Michel Marcus and Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 545 terms from Michel Marcus)

Crossrefs

Cf. A001065, A037020, A072326 (dead).

Programs

Mathematica

Table[SelectFirst[Range[10^7], PrimeQ[DivisorSigma[1, #] - #] &[# n] &] /. k_ /; MissingQ@ k -> -1, {n, 77}] (* Michael De Vlieger, Aug 01 2017 *)

PARI

a(n) = {my(k=1); while (!isprime(sigma(k*n)-k*n), k++); k;}

Formula

a(A037020(n)) = 1.