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 A269781 a(n) is the smallest k different from n such that (n, k) is an amicably refactorable pair (see the comments). 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 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. LINKS EXAMPLE 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. 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 CROSSREFS Cf. A000005 (number of divisors), A033950 (refactorable numbers), A061286 (subsequence for odd prime indices and semiprime indices), A268037, A272353 (related sequences). Sequence in context: A271011 A077215 A270908 * A322144 A272580 A065679 Adjacent sequences:  A269778 A269779 A269780 * A269782 A269783 A269784 KEYWORD nonn AUTHOR Waldemar Puszkarz, May 01 2016 STATUS approved

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Last modified August 11 15:12 EDT 2020. Contains 336428 sequences. (Running on oeis4.)