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

1,2

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Sequence in context: A326947 A256450 A119605 * A284763 A090240 A137407

Adjacent sequences: A144143 A144144 A144145 * A144147 A144148 A144149

Keyword

nonn

Author

Leroy Quet, Sep 11 2008

Extensions

More terms from Stefan Steinerberger, Sep 15 2008

Status

approved