A269781 a(n) is the smallest k different from n such that (n, k) is an amicably refactorable pair (see the comments).

4, 3, 16, 4, 64, 24, 36, 16, 1024, 6, 4096, 64, 4, 5, 65536, 12, 262144, 6, 4, 1024, 4194304, 8, 81, 4096, 4, 6, 268435456, 16, 1073741824, 6, 4, 65536, 16, 9, 68719476736, 262144, 4, 8, 1099511627776, 32, 4398046511104, 6, 36, 4194304, 70368744177664, 10, 729, 48, 4, 6, 4503599627370496, 32, 16, 8

Offset: 3

Author

Waldemar Puszkarz, May 01 2016

Keywords

• nonn

Comments

Let m and k be distinct integers and numdiv(n) be the number of divisors of n (A000005(n)). We call m and k amicably refactorable if numdiv(m) divides k and numdiv(k) divides m.

For any n with no amicably refactorable partner, a(n) = 0.

Conjecture: the sequence contains no zeros.

1 does not have an amicable partner as all other numbers have more than one divisor and 2 does not have an amicable partner as all other numbers with two divisors are odd primes and cannot be divided by the number of divisors of 2, also 2. All other numbers may have an amicably refactorable partner, though for some, primes, semiprimes and squares of primes in particular, this number can be quite large.

For primes and semiprimes, a(n) = 2^(f(n) - 1), (see A061286), where f(n) is their largest prime factor. For squares of primes, a(n) = 3^(|sqrt(n)| - 1), except for n = 9 for which this formula yields 9; this forces us to choose the next best candidate: 36.

Examples

			For n=5, a(5)=16 as the number of divisors of n (2) divides a(n) while the number of divisors of a(n) (5) divides 5 and 16 is the smallest number for which this happens.
		

Crossrefs

Cf. A000005 (number of divisors), A033950 (refactorable numbers), A061286 (subsequence for odd prime indices and semiprime indices), A268037, A272353 (related sequences).

Programs

Mathematica

A269781 = {}; Do[k = 1; If[PrimeQ[n] || PrimeNu[n] == 2 && PrimeOmega[n] == 2, AppendTo[A269781, 2^(First[Last[FactorInteger[n]]] - 1)], If[PrimeQ @ Sqrt @ n && (n > 9), AppendTo[A269781, 3^(Sqrt[n] - 1)],While[k != n && !(Divisible[n, DivisorSigma[0, k]] && Divisible[k, DivisorSigma[0, n]]), k++]; If[k == n, k = n + 1; While[!(Divisible[n, DivisorSigma[0, k]] && Divisible[k, DivisorSigma[0, n]]), k++]]; AppendTo[A269781, k]]], {n, 3, 56}]; A269781