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A347356 a(n) = m/A006939(A001221(m)) with m = A347284(n).

%I A347356 #9 Oct 16 2021 06:14:46
%S A347356 1,1,2,2,2,4,24,24,48,48,96,576,576,1152,34560,207360,414720,414720,
%T A347356 829440,174182400,1045094400,2090188800,2090188800,2090188800,
%U A347356 4180377600,25082265600,25082265600,50164531200,1504935936000,3009871872000,18059231232000
%N A347356 a(n) = m/A006939(A001221(m)) with m = A347284(n).
%H A347356 Michael De Vlieger, <a href="/A347356/b347356.txt">Table of n, a(n) for n = 1..581</a>
%H A347356 Michael De Vlieger, <a href="/A347356/a347356.png">Diagram of prime power decompositions of A347284(n)</a> for 2 <= n <= 14, showing a(n) in blue, A006939(A089576(n)) in red, and the number A347284(n) in black.
%F A347356 a(n) = A347284(n)/A006939(A089576(n)).
%e A347356 Diagram of prime power decomposition of A347284(12) = 2^12 * 3^7 * 5^4 * 7^3 * 11^2 * 13, showing Chernoff number A006939(6) with "x" and "X", A002110(A347354(12)) with "X", and a(12) with "o":
%e A347356           12  o
%e A347356           11  o
%e A347356           10  o
%e A347356            9  o
%e A347356            8  o
%e A347356            7  o o
%e A347356            6  x o
%e A347356            5  x x
%e A347356            4  x x x
%e A347356            3  x x x x
%e A347356            2  x x x x x
%e A347356            1  X X X X X X
%e A347356               2 3 5 7 ...
%e A347356 A347284(12) = A006939(6) * a(12)
%e A347356           = 5244319080000 * 576
%e A347356           = 3020727790080000.
%t A347356 Block[{nn = 31, a = {}, b, c, e, i, p}, Array[Set[e[#], 0] &, Floor[2^# If[# <= 4, 1/2, -1 + 2^(7/(3 #))]] &[Ceiling@ Log2@ nn]]; Do[e[1]++; b = {2^e[1]}; c = {e[1]}; Do[If[Last[b] == 1, Break[], i = e[j]; p = Prime[j]; While[p^i < b[[j - 1]], i++]; AppendTo[b, p^(i - 1)]; AppendTo[c, (i - 1)]; If[i > e[j], e[j]++]], {j, 2, k}]; AppendTo[a, If[First[#] == 0, 1, Times @@ MapIndexed[Prime[First[#2]]^#1 &, TakeWhile[#, # > 0 &]]] &[# - Range[Length[#], 1, -1]] &@ If[k > 2, Most@ c, c]], {k, nn}]; a]
%Y A347356 Cf. A001221, A006939, A089576, A347284.
%K A347356 nonn
%O A347356 1,3
%A A347356 _Michael De Vlieger_, Oct 02 2021