A124184: prime factorization of 1?

[franktaw at netscape.net]

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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