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

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, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 49, 51, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95

Offset: 1

Author

Leroy Quet, Sep 11 2008

Keywords

• nonn

Comments

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.

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Programs

Maple

filter:= proc(n) local E;
  E:= map(t -> t[2], ifactors(n)[2]);
  andmap(t -> igcd(t,n)=1, E)
end proc:
select(filter, [$1..200]); # Robert Israel, Oct 24 2019

Mathematica

Select[Range[100], GCD[Times @@ Table[FactorInteger[ # ][[i, 2]], {i, 1, Length[FactorInteger[ # ]]}], # ] == 1 &] (* Stefan Steinerberger, Sep 15 2008 *)

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

Extensions

More terms from Stefan Steinerberger, Sep 15 2008