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A345882 Number of numbers expressible as b(1)*b(2)*...*b(n) with 1 <= b(i) <= i for each i. 4
1, 2, 5, 11, 30, 64, 178, 382, 758, 1367, 3620, 7193, 19707, 40867, 75706, 130017, 339506, 667390, 1824656, 3724917, 6785689, 11545898, 30099090, 58833294, 105348580, 176098677, 282847446, 438090287, 1095200628, 2057512312, 5494259815, 10925293558, 19311381148 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

On replacing * with +, one gets A000124.

In other words, take n! = 1*2*3*...*n and replace any factor by any smaller number. a(n) is the number of different numbers that can be obtained. If b(i) is required to be a divisor of i, we get A027423. - N. J. A. Sloane, Sep 18 2021

LINKS

Table of n, a(n) for n=1..33.

EXAMPLE

For n=3, b(1) must equal 1, b(2) can be 1 or 2, and b(3) can be 1, 2 or 3. This gives 3!=6 possible products: 1*1*1=1, 1*2*1=2, 1*1*2=2, 1*1*3=3, 1*2*2=4 and 1*2*3=6. Since 1*2*1=1*1*2, this process yields 5 distinct numbers, so a(3)=5.

MATHEMATICA

list[1] := {1};

list[n_] := list[n] = DeleteDuplicates[Flatten[Table[i*list[n - 1], {i, 1, n}]]];

a[n_] := a[n] = Length[list[n]]; Table[a[n], {n, 1, 10}]

PROG

(PARI) a(n) = my(l = List()); forvec(x = vector(n, i, [1, i]), listput(l, prod(i = 1, n, x[i])), 1); listsort(l, 1); #l \\ David A. Corneth, Sep 18 2021

(Python)

def A345882set(n):

    if n == 1:

        return {1}

    else:

        s = A345882set(n-1)

        c = set(s)

        for x in s:

            for i in range(2, n+1):

                c.add(i*x)

        return c

def A345882(n): return len(A345882set(n)) # Chai Wah Wu, Sep 19 2021

CROSSREFS

Cf. A000124, A027423, A110713, A347685 (first differences), A347686.

Sequence in context: A176637 A272675 A059075 * A121134 A144959 A131347

Adjacent sequences:  A345879 A345880 A345881 * A345883 A345884 A345885

KEYWORD

nonn,nice

AUTHOR

David Galvin, Sep 16 2021

EXTENSIONS

a(26)-a(28) from Chai Wah Wu, Sep 19 2021

a(29)-a(33) from Martin Ehrenstein, Sep 22 2021

STATUS

approved

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Last modified October 2 08:07 EDT 2022. Contains 357191 sequences. (Running on oeis4.)