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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.
10

%I #16 Sep 29 2024 02:42:09

%S 1,4,8,16,32,64,128,144,216,288,432,864,1152,1296,1728,2048,2592,3456,

%T 5184,7776,8192,10368,13824,15552,18432,20736,31104,41472,55296,62208,

%U 73728,86400,108000,129600,131072,165888,194400,216000,221184,259200,279936,324000

%N 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 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 Amiram Eldar, <a href="/A353500/b353500.txt">Table of n, a(n) for n = 1..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Highly_powerful_number">Highly powerful number</a>.

%e 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 nn=1000;

%t d=Table[Times@@Last/@FactorInteger[n],{n,nn}];

%t Select[Range[nn],!MemberQ[Take[d,#-1],d[[#]]]&]

%t 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 These are the positions of first appearances in A005361, counted by A266477.

%Y This is the sorted version of A085629.

%Y The version for shadows instead of exponents is A353397, firsts in A353394.

%Y A001222 counts prime factors with multiplicity, distinct A001221.

%Y A003963 gives product of prime indices, counted by A339095.

%Y A056239 adds up prime indices, row sums of A112798 and A296150.

%Y A124010 gives prime exponents, sorted A118914.

%Y A130091 lists numbers with distinct prime exponents, counted by A098859.

%Y A181819 gives prime shadow, with an inverse A181821.

%Y Cf. A070175, A097318, A116608, A162247, A182850, A304678, A325131, A325238, A353399, A353503, A353506, A353507.

%Y Subsequence of A181800.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 17 2022