A304938 a(n) is the smallest number which can be written in n different ways as an ordered product of prime factors.

1, 6, 12, 24, 48, 30, 192, 384, 768, 72, 3072, 60, 12288, 24576, 144, 98304, 196608, 393216, 786432, 120, 288, 6291456, 12582912, 210, 50331648, 100663296, 201326592, 576, 805306368, 180, 3221225472, 6442450944, 12884901888, 25769803776, 432, 1152, 206158430208, 412316860416, 824633720832, 1649267441664

Offset

1,2

Comments

It can be easily demonstrated that a(n) exists for all n and is less than or equal to 2^(n-1)*3 since 2^(n-1)*3 can be written in n different ways.

If n is a prime number then a(n) = 2^(n-1)*3, but there are also nonprime numbers n with this property (e.g., 9 and 14).

If n=k! then a(n) is the product of the first k prime numbers.

Finding the terms up to 2^64 was the focus of the 4th question of the ICPC coding challenge in 2013.

Links

Amiram Eldar, Table of n, a(n) for n = 1..136

The ACM-ICPC International Collegiate Programming Contest, ICPC 2013 problems

Formula

a(p) = 2^(p-1)*3 if p is a prime.

a(k!) = prime(k)# is the k-th primorial number. So for no m < k!, prime(k) | a(m). - David A. Corneth, May 24 2018

a(n) = min { k : A008480(k) = n }. - Alois P. Heinz, May 26 2018

Example

a(1) = 1 because only a prime power or the empty product (which equals 1) can be written in just one way, and no prime power is smaller than 1.
a(2) = 6 = 3 * 2 = 2 * 3 because none of 3, 4, 5 can be written in two different ways.
a(3) = 12 = 3 * 2 * 2 = 2 * 3 * 2 = 2 * 2 * 3 (each of 7, 8, 9, 10, 11 can be written in at most 2 ways).
a(4) = 24 = 2 * 2 * 2 * 3 (each of 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 can be written in at most 3 ways).

Mathematica

uv=Table[Length[Permutations[Join@@ConstantArray@@@FactorInteger[n]]], {n, 1, 1000}];
Table[Position[uv, k][[1, 1]], {k, Min@@Complement[Range[Max@@uv], uv]-1}] (* Gus Wiseman, Nov 22 2022 *)

Prog

(PARI) a008480(n) = my(f=factor(n)); sum(k=1, #f~, f[k, 2])!/prod(k=1, #f~, f[k, 2]!);
a(n) = {my(k=2); while (a008480(k) !=n, k++); k; } \\ Michel Marcus, May 23 2018

Crossrefs

Cf. A000961, A007283 (2^n*3), A008480 (number of ordered prime factorizations of n).

Subsequence of A025487.

The sorted version is A358526.

Cf. A001222, A027746, A206487, A344606, A358508.

Sequence in context: A199910 A210678 A358508 * A297107 A330997 A331200

Adjacent sequences: A304935 A304936 A304937 * A304939 A304940 A304941

Keyword

nonn

Author

Vincent Champain, May 21 2018

Status

approved