一个OEIS序列的问题

northwolves

今天OEIS投了个稿，编辑说似乎跟某个老序列均差1。大家帮忙看看是不是有差异？

A343975
NAME       
The least sum of most number of factors which are different and their product is equal to n.

DATA       
1, 3, 4, 5, 6, 6, 8, 7, 10, 8, 12, 8, 14, 10, 9, 11, 18, 10, 20, 10

OFFSET       
1,2

EXAMPLE       
For n = 12, its divisors are [1, 2, 3, 4, 6, 12]. We can express 12 as product of most 3 factors: 1*2*6 or 1*3*4. 1+3+4=8<1+2+6, thus a(12) = 8.

For n = 18, its divisors are [1,2,3,6,9,18]. We can express 18 as product of most 3 factors: 1*2*9 or 1*3*6. 1+3+6=10<1+2+9, thus a(18) = 10.

-------------------------------------------------------------
       
Kevin Ryde: A319057 + 1 looks same for many terms, but the definition here and there are not quite the same. Will it differ eventually? (If not then you may like to extend A319057 instead of a new sequence.)

-------------------------------------------------------------
A319057                Minimum sum of a strict factorization of n into factors > 1.               

0, 2, 3, 4, 5, 5, 7, 6, 9, 7, 11, 7, 13, 9, 8, 10, 17, 9, 19, 9, 10, 13, 23, 9, 25, 15, 12, 11, 29, 10, 31, 12, 14, 19, 12, 11, 37, 21, 16, 11, 41, 12, 43, 15, 14, 25, 47, 12, 49, 15, 20, 17, 53, 14, 16, 13, 22, 31, 59, 12, 61, 33, 16, 14, 18, 16, 67, 21, 26 (list; graph; refs; listen; history; edit; text; internal format)
OFFSET       
1,2

COMMENTS       
a(n) >= A001414(n), with equality iff n is squarefree or four times a squarefree number (i.e., A000188(n) <= 2). - Charlie Neder, Sep 10 2018

LINKS       
Charlie Neder, Table of n, a(n) for n = 1..1000

EXAMPLE       
The strict factorizations of 48 are (48), (2*24), (3*16), (4*12), (6*8), (2*3*8), (2*4*6), with sums 48, 26, 19, 16, 14, 13, 12 respectively, so a(48) = 12.

MATHEMATICA       
strfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#, d]&)/@Select[strfacs[n/d], Min@@#>d&], {d, Rest[Divisors[n]]}]];

Table[Min[Total/@strfacs[n]], {n, 100}]

CROSSREFS       
Cf. A001055, A007716, A045778, A056239, A162247, A215366, A246868, A318871, A318953, A318954.

Sequence in context: A134875 A134889 A303702 * A181894 A265535 A094802

Adjacent sequences:  A319054 A319055 A319056 * A319058 A319059 A319060

KEYWORD       
nonn

AUTHOR       
Gus Wiseman, Sep 09 2018

STATUS       
approved

northwolves

百度翻译

The least sum of most number of factors which are different and their product is equal to n.

各因子之积等于n的多数因子的最小和。

Minimum sum of a strict factorization of n into factors > 1.

n到因子>1的严格因子分解的最小和。

mathe

两者定义的唯一区别是前一个包含了因子1，后者不包含因子1吧？
当然前者还加了额外的数目最多的要求，但是和最小同样意味着数目最多，所以这个条件可以省略

northwolves

从题目看Minimum sum of a strict factorization of n into factors简练多了

mathematica

northwolves 发表于 2021-5-7 08:02
从题目看Minimum sum of a strict factorization of n into factors简练多了

你应该翻译一下，表达一下你的数列是啥意思

northwolves

mathematica 发表于 2021-5-7 08:06
你应该翻译一下，表达一下你的数列是啥意思

我的初步想法就是把n表示成某几个因数的积，因数个数最多的情况下其和值最小。

王守恩

mathematica 发表于 2021-5-7 08:06
你应该翻译一下，表达一下你的数列是啥意思

题目看不懂，但好像是这样：每个a(n)不会比 n 大，每个 n 都会在a(n)中出现

A319057(挺好玩的：把n表示成不同因数(不能是1)的积)
{0, 2, 3, 4,
5, 5, 7, 6, 9,
7, 11, 7, 13, 9, 8, 10,
17, 9, 19, 9, 10, 13, 23, 9, 25,
15, 12, 11, 29, 10, 31, 12, 14, 19, 12, 11,
37, 21, 16, 11, 41, 12, 43, 15, 14, 25, 47, 12, 49,
15, 20, 17, 53, 14, 16, 13, 22, 31, 59, 12, 61, 33, 16, 14,
18, 16, 67, 21, 26, 14, 71, 13, 73, 39, 20, 23, 18, 18, 79, 15, 30,
43, 83, 14, 22, 45, 32, 17, 89, 14, 20, 27, 34, 49, 24, 15, 97, 21, 20, 17,
101, 22, 103, 19, 15, 55, 107, 16, 109, 18, 40, 17, 113, 24, 28, 33, 22, 61, 24, 14, 121,
63, 44, 35, 30, 16, 127, 22, 46, 20, 131, 18, 26, 69, 17, 23, 137, 28, 139, 16, 50, 73, 24, 15,
34, 75, 28, 41, 149, 18, 151, 25, 26, 20, 36, 20, 157, 81, 56, 17, 30, 18, 163, 45, 19, 85, 167, 16, 169,
24, 28, 47, 173, 34, 32, 21, 62, 91, 179, 16, 181, 22, 64, 29, 42, 36, 28, 51, 19, 26, 191, 17, 193, 99, 21, 23,
197, 20, 199, 19, 70, 103, 36, 24, 46, 105, 32, 23, 30, 17, 211, 57, 74, 109, 48, 18, 38, 111, 76, 20, 30, 42, 223, 19, 23,
115, 227, 26, 229, 30, 21, 35, 233, 22, 52, 63, 82, 26, 239, 17, 241, 33, 36, 65, 42, 46, 32, 37, 86, 32, 251, 18, 34, 129, 25, 26,
257, 48, 44, 22, 38, 133, 263, 20, 58, 28, 92, 71, 269, 19, 271, 27, 23, 139, 36, 30, 277, 141, 40, 18, 281, 52, 283, 75, 27, 26, 48, 19, 289}
应该是这样：把n表示成不同因数(因数不能是1)的积
譬如：36=2*2*3*3(相同的不行)=3*3*4(相同的不行)=2*3*6=11
譬如：81=3*3*3*3(相同的不行)=3*3*9(相同的不行)=3*27=30
譬如：120=2*3*4*5=14
譬如：144=2*3*4*6=15
譬如：240=2*4*5*6=17
譬如：720=2*3*4*5*6=20

王守恩

northwolves 发表于 2021-5-7 09:09
我的初步想法就是把n表示成某几个因数的积，因数个数最多的情况下其和值最小。

A001414:把 n 表示成某几个因数的积，其和值最小。A001414 比A319057 还是简单些。       
{0, 2, 3, 4,
5, 5, 7, 6, 6,
7, 11, 7, 13, 9, 8, 8,
17, 8, 19, 9, 10, 13, 23, 9, 10,
15, 9, 11, 29, 10, 31, 10, 14, 19, 12, 10,
37, 21, 16, 11, 41, 12, 43, 15, 11, 25, 47, 11, 14,
12, 20, 17, 53, 11, 16, 13, 22, 31, 59, 12, 61, 33, 13, 12,
18, 16, 67, 21, 26, 14, 71, 12, 73, 39, 13, 23, 18, 18, 79, 13, 12,
43, 83, 14, 22, 45, 32, 17, 89, 13, 20, 27, 34, 49, 24, 13, 97, 16, 17, 14,
101, 22, 103, 19, 15, 55, 107, 13, 109, 18, 40, 15, 113, 24, 28, 33, 19, 61, 24, 14, 22,
63, 44, 35, 15, 15, 127, 14, 46, 20, 131, 18, 26, 69, 14, 23, 137, 28, 139, 16, 50, 73, 24, 14,
34, 75, 17, 41, 149, 15, 151, 25, 23, 20, 36, 20, 157, 81, 56, 15, 30, 14, 163, 45, 19, 85, 167, 16, 26,
24, 25, 47, 173, 34, 17, 19, 62, 91, 179, 15, 181, 22, 64, 29, 42, 36, 28, 51, 16, 26, 191, 15, 193, 99, 21, 18,
197, 19, 199, 16, 70, 103, 36, 24, 46, 105, 29, 21, 30, 17, 211, 57, 74, 109, 48, 15, 38, 111, 76, 20, 30, 42, 223, 17, 16,
115, 227, 26, 229, 30, 21, 35, 233, 21, 52, 63, 82, 26, 239, 16, 241, 24, 15, 65, 19, 46, 32, 37, 86, 17, 251, 17, 34, 129, 25, 16,
257, 48, 44, 22, 35, 133, 263, 20, 58, 28, 92, 71, 269, 16, 271, 25, 23, 139, 21, 30, 277, 141, 37, 18, 281, 52, 283, 75, 27, 26, 48, 16, 34}