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A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

%I A353500 #16 Sep 29 2024 02:42:09
%S A353500 1,4,8,16,32,64,128,144,216,288,432,864,1152,1296,1728,2048,2592,3456,
%T A353500 5184,7776,8192,10368,13824,15552,18432,20736,31104,41472,55296,62208,
%U A353500 73728,86400,108000,129600,131072,165888,194400,216000,221184,259200,279936,324000
%N A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.
%C A353500 All terms are highly powerful (A005934), but that sequence looks only at first appearances that reach a record, and is missing 1152, 2048, 8192, etc.
%H A353500 Amiram Eldar, <a href="/A353500/b353500.txt">Table of n, a(n) for n = 1..10000</a>
%H A353500 Wikipedia, <a href="https://en.wikipedia.org/wiki/Highly_powerful_number">Highly powerful number</a>.
%e A353500 The prime exponents of 86400 are (7,3,2), and this is the first case of product 42, so 86400 is in the sequence.
%t A353500 nn=1000;
%t A353500 d=Table[Times@@Last/@FactorInteger[n],{n,nn}];
%t A353500 Select[Range[nn],!MemberQ[Take[d,#-1],d[[#]]]&]
%t A353500 lps[fct_] := Module[{nf = Length[fct]}, Times @@ (Prime[Range[nf]]^Reverse[fct])]; lps[{1}] = 1; q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, (n == 1 || AllTrue[e, # > 1 &]) && n == Min[lps /@ f[Times @@ e]]]; Select[Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {_, _}][[;; , 2]], q] (* _Amiram Eldar_, Sep 29 2024, using the function f by T. D. Noe at A162247 *)
%Y A353500 These are the positions of first appearances in A005361, counted by A266477.
%Y A353500 This is the sorted version of A085629.
%Y A353500 The version for shadows instead of exponents is A353397, firsts in A353394.
%Y A353500 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A353500 A003963 gives product of prime indices, counted by A339095.
%Y A353500 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353500 A124010 gives prime exponents, sorted A118914.
%Y A353500 A130091 lists numbers with distinct prime exponents, counted by A098859.
%Y A353500 A181819 gives prime shadow, with an inverse A181821.
%Y A353500 Cf. A070175, A097318, A116608, A162247, A182850, A304678, A325131, A325238, A353399, A353503, A353506, A353507.
%Y A353500 Subsequence of A181800.
%K A353500 nonn
%O A353500 1,2
%A A353500 _Gus Wiseman_, May 17 2022