A363501 a(n) = smallest product > n of some subset of the divisors of n, or if no product > n exists then a(n) = n.

1, 2, 3, 8, 5, 12, 7, 16, 27, 20, 11, 18, 13, 28, 45, 32, 17, 27, 19, 40, 63, 44, 23, 32, 125, 52, 81, 56, 29, 36, 31, 64, 99, 68, 175, 48, 37, 76, 117, 50, 41, 63, 43, 88, 75, 92, 47, 64, 343, 100, 153, 104, 53, 81, 275, 64, 171, 116, 59, 72, 61, 124, 147

Offset: 1

Author

Denis Ivanov, Jun 06 2023

Keywords

• nonn

Comments

a(n) = n iff n=1 or n is prime.

For composite n, A363627(n) < n < a(n) and where both bounds are products of divisors of n and as tight as possible.

Examples

			n = 4; divisors: [1,2,4]; subsets: [[], [1], [2], [4], [1, 2], [1, 4], [2, 4], [1, 2, 4]]; products: [1, 1, 2, 4, 2, 4, 8, 8]; minimal product that greater than 4 is 8, so a(4) = 8.
n = 5; divisors: [1,5]; subsets: [[], [1], [5], [1, 5]]; products: [1, 1, 5, 5]; no products greater than 5, so a(5) = 5.
		

Crossrefs

Cf. A027750, A363627.

Programs

Mathematica

a[n_] := If[PrimeQ@n || n == 1, n,
  First@Select[Union[Times @@@ Subsets[Divisors@n]], # > n &]];

PARI

a(n) = my(d=divisors(n), nb = #d, m=oo); forsubset(nb, s, my(p=vecprod(vector(#s, k, d[s[k]]))); if (p>n, m=min(m, p))); if (mMichel Marcus, Jun 17 2023