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A144146 A positive integer k is included if every nonzero exponent in the prime factorization of k is coprime to k.

%I A144146 #17 Feb 11 2024 02:40:08
%S A144146 1,2,3,5,6,7,8,9,10,11,13,14,15,17,19,21,22,23,25,26,29,30,31,32,33,
%T A144146 34,35,37,38,39,40,41,42,43,45,46,47,49,51,53,55,56,57,58,59,61,62,63,
%U A144146 65,66,67,69,70,71,73,74,75,77,78,79,81,82,83,85,86,87,88,89,91,93,94,95
%N A144146 A positive integer k is included if every nonzero exponent in the prime factorization of k is coprime to k.
%C A144146 1 is included somewhat arbitrarily. 1 has no nonzero exponents in its prime factorization, but it also has no prime factorization exponents that are not coprime to 1.
%C A144146 The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 9, 75, 746, 7433, 74270, 742714, 7427050, 74270567, 742705640, 7427055214, ... . Apparently, the asymptotic density of this sequence exists and equals 0.742705... .  - _Amiram Eldar_, Feb 11 2024
%H A144146 Robert Israel, <a href="/A144146/b144146.txt">Table of n, a(n) for n = 1..10000</a>
%e A144146 40 has the prime-factorization 2^3 * 5^1. The exponents are therefore 3 and 1. Since both 3 and 1 are coprime to 40, then 40 is included in the sequence.
%p A144146 filter:= proc(n) local E;
%p A144146   E:= map(t -> t[2], ifactors(n)[2]);
%p A144146   andmap(t -> igcd(t,n)=1, E)
%p A144146 end proc:
%p A144146 select(filter, [$1..200]); # _Robert Israel_, Oct 24 2019
%t A144146 Select[Range[100], GCD[Times @@ Table[FactorInteger[ # ][[i, 2]], {i, 1, Length[FactorInteger[ # ]]}], # ] == 1 &] (* _Stefan Steinerberger_, Sep 15 2008 *)
%o A144146 (PARI) is(n) = {my(e = factor(n)[, 2]); for(i=1, #e, if(gcd(e[i], n) > 1, return(0))); 1;}; \\ _Amiram Eldar_, Feb 11 2024
%K A144146 nonn
%O A144146 1,2
%A A144146 _Leroy Quet_, Sep 11 2008
%E A144146 More terms from _Stefan Steinerberger_, Sep 15 2008