A353500 - OEIS

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.

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1152, 1296, 1728, 2048, 2592, 3456, 5184, 7776, 8192, 10368, 13824, 15552, 18432, 20736, 31104, 41472, 55296, 62208, 73728, 86400, 108000, 129600, 131072, 165888, 194400, 216000, 221184, 259200, 279936, 324000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Wikipedia, Highly powerful number.

EXAMPLE

The prime exponents of 86400 are (7,3,2), and this is the first case of product 42, so 86400 is in the sequence.

MATHEMATICA

nn=1000;

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

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

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 *)

CROSSREFS

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

This is the sorted version of A085629.

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

A001222 counts prime factors with multiplicity, distinct A001221.

A003963 gives product of prime indices, counted by A339095.

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

A124010 gives prime exponents, sorted A118914.

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

A181819 gives prime shadow, with an inverse A181821.