A124184: prime factorization of 1?

[Jonathan Post]

Excellent comments, Frank!  I hope these are added by njas to Zak's (in my
opinion) nice sequence.

-- Jonathan Vos Post

On 12/7/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
> I can confirm Zak's values.  (Except that 1 really does belong in the
> sequence.)
>
> We really also should have the sequence of the numbers that are not
> divisible by at least one of the (positive) exponents in their prime
> factorization:
>
> 8,9,25,32,40,45,49,56,63,64,75,81,88,96,99,104,117,121,125,128,136,147,
> 152,153,162,169,171,175,184,200,207,224,225,232,243,245,248,250,261,275,
> 279,280,288,289,296,315,320,325,328,333,343,344,352,361,363,369,376,384,
> 387,392,405,416,423,424,425,440,441,448,472,475,477,486,488,495,500
>
> In this case, 1 does not belong; it has no prime exponents, so there
> aren't any of them that fail to divide it.
>
> Also, in this case we can't get around it by talking about 0 exponents;
> if we included these, the sequence would be the natural numbers.
>
> Except for 1, every member of A124184 is in this sequence.  Also, this
> sequence does not contain any square-free numbers.
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: zakseidov at yahoo.com
>
> Dear seqfans,
>
> In A124184,
> shouldn't we omit 1
> (I omit it, while Leroy includes)?
>
> What is prime factorization of 1?
>
> Mmca gives:
>
> FactorInteger[1]
> {}
>
> Wiki says:
> 1 has no prime factors. (1 is the empty product)
>
> Also, is there any special name of numbers with
> all  prime-factorization exponents >1 ?
>
> In A124184, we have only such numbers!
>
> Thanks, Zak
>
>
> (***************************************)
> %I A124184
> %S A124184
> 8,9,25,32,49,64,81,121,125,128,169,225,243,289,343,361,441,512,529,625,72
> 9,841,961,1000,1024,1089,1225,1331,1369,1521,1681,1849,2025,2048,2187,219
> 7,2209,2401,2601,2744,2809,3025,3249,3481,3721,3969,4096,4225,4489,4761,4
> 913,5041,5329,5625,5929,6125,6241,6561,6859,6889,7225,7569,7776,7921,8000
> .
> %K A124184 nonn,new
> %O A124184 1,2
> %D A124184 More terms from Zak Seidov
> (zakseidov(AT)yahoo.com), Dec 08 2006
>
>
>
>
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