# [seqfan] Re: nonpowers in A001481

Charles Greathouse charles.greathouse at case.edu
Sun Jan 16 18:09:11 CET 2011

```Not quite -- this disallows 5^3 * 13.  What about

Numbers where no prime of the form 4k+3 appears with an odd exponent
in the prime factorization of the number, and where the gcd of the
exponents is 1.

Here's Pari code:
isA180161(n)=my(f=factor(n),g=0);for(i=1,#f[,2],if(f[i,1]%4==3&f[i,2]%2,return(0));g=gcd(g,f[i,2]));g==1&n>1

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Sun, Jan 16, 2011 at 12:00 PM, Joerg Arndt <arndt at jjj.de> wrote:
> Me stupid.  Mentally took zero-exponents as ones:
>
> How about rewording the current
>
>  These are the numbers n = 2^i Product_{p == 3 mod 4} p^2j Product_{p
>  == 1 mod 4 } p^k where gcd of the nonzero exponents {i, 2j's, k's} is 1.
>
> to
>
>  Numbers of the form 2^i * S^j * T^(2*k) where S is a product of
>  primes 4k+1, T a product of primes 4k+3, and i,j,k>=0, and
>  gcd(i, j, 2*k)==1.
>
> ?
>
> ...hoping the latter is correct as well.
>
>
> * Charles Greathouse <charles.greathouse at case.edu> [Jan 16. 2011 17:42]:
>> But for 9,  gcd{i, 2j's, k's} = gcd {2} = 2 ≠ 1, so it shouldn't be a
>> member.  For 18, gcd{i, 2j's, k's} = gcd {1, 2} = 1 so it should be a
>> member.
>>
>> Charles Greathouse
>> Analyst/Programmer
>> Case Western Reserve University
>>
>> [...]
>
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>

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