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A345882 Number of numbers expressible as b(1)*b(2)*...*b(n) with 1 <= b(i) <= i for each i.

%I A345882 #56 Sep 22 2021 04:58:01
%S A345882 1,2,5,11,30,64,178,382,758,1367,3620,7193,19707,40867,75706,130017,
%T A345882 339506,667390,1824656,3724917,6785689,11545898,30099090,58833294,
%U A345882 105348580,176098677,282847446,438090287,1095200628,2057512312,5494259815,10925293558,19311381148
%N A345882 Number of numbers expressible as b(1)*b(2)*...*b(n) with 1 <= b(i) <= i for each i.
%C A345882 On replacing * with +, one gets A000124.
%C A345882 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
%e A345882 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.
%t A345882 list[1] := {1};
%t A345882 list[n_] := list[n] = DeleteDuplicates[Flatten[Table[i*list[n - 1], {i, 1, n}]]];
%t A345882 a[n_] := a[n] = Length[list[n]]; Table[a[n], {n, 1, 10}]
%o A345882 (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
%o A345882 (Python)
%o A345882 def A345882set(n):
%o A345882     if n == 1:
%o A345882         return {1}
%o A345882     else:
%o A345882         s = A345882set(n-1)
%o A345882         c = set(s)
%o A345882         for x in s:
%o A345882             for i in range(2,n+1):
%o A345882                 c.add(i*x)
%o A345882         return c
%o A345882 def A345882(n): return len(A345882set(n)) # _Chai Wah Wu_, Sep 19 2021
%Y A345882 Cf. A000124, A027423, A110713, A347685 (first differences), A347686.
%K A345882 nonn,nice
%O A345882 1,2
%A A345882 _David Galvin_, Sep 16 2021
%E A345882 a(26)-a(28) from _Chai Wah Wu_, Sep 19 2021
%E A345882 a(29)-a(33) from _Martin Ehrenstein_, Sep 22 2021