<Text-field style="Heading 1" layout="Heading 1">Worksheet Information</Text-field>

<Text-field style="Heading 1" layout="Heading 1">A Counter-Example on the Density of Integers</Text-field>

The following problem appeared in the Puzzles Column of the Fall 2004 Emissary, newsletter of the MSRI. The Puzzles Columns is edited by Elwyn Berlekamp and Joe Buhler. See also http://www.msri.org/communications/emissary/pdfs/EmissaryFall04.pdf.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 2" layout="Heading 2">The problem</Text-field>

A set SSJBRzYi of positive integers is said to have density SSJkRzYi if the fraction of integers in NyQiIiJJIm5HNiI= that are in SSJBRzYi approaches SSJkRzYi as SSJuRzYi goes to infinity, i.e.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JlEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjFRMSZJbnZpc2libGVUaW1lcztGJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRlIvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRjk2MVEiPUYnRjJGPEY+RkFGQ0ZFRkdGSUZLRk1GUEZTRlVGWEY4LUklbXN1YkdGJDYlLUYsNiZRJGxpbUYnL0YwRkBGMkY8LUYjNiUtRiw2JlEibkYnRi9GMkY1LUY5NjFRLSZSaWdodEFycm93O0YnRjJGPEY+RkEvRkRGMUZFRkdGSUZLRk1GUEZTRlVGWC1JI21uR0YkNiVRKCZpbmZpbjtGJ0YyRjwvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLDYmUSFGJ0Zeb0YyRjwtSSZtZnJhY0dGJDYoLUYjNiUtRiw2JkZhcEYvRjJGNS1GIzYlLUYsNiZRIkFGJ0YvRjJGNS1GOTYxUTAmQXBwbHlGdW5jdGlvbjtGJ0YyRjxGPkZBRkNGRUZHRklGS0ZNL0ZRUSQwZW1GJy9GVEZicUZVRlgtSShtZmVuY2VkR0YkNiMtRiM2I0Zhb0ZncEZhby8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGXnIvJSliZXZlbGxlZEdGQEZfcC1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC4wZW1GJy8lJmRlcHRoR0Zoci8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYlRmFwRi9GNQ==where LUkiQUc2IjYjSSJuR0Ykis the cardinality of the set of elements of SSJBRzYi of size at most 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.

Find two sets of density IyIiIiIiIw== whose intersection does not have a density, i.e. the relevant limit does not exist.

<Text-field style="Heading 2" layout="Heading 2">The solution</Text-field>

In the following, the interval notation Oy1JJU9wZW5HNiI2I0kiSkdGJUkiS0dGJQ== indicates the integers x satisfying J < x\342\211\244 K, the interval notation [ J, K ] indicates the integers x satisfying J \342\211\244 x \342\211\244 K and so on.

<Text-field style="Heading 3" italic="false" layout="Heading 3">The Set A and its density</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Let A be the set of odd integers. Then the density of A is IyIiIiIiIw==. This follows because if n is even, then there are LCRJIm5HNiIjIiIiIiIj

odd integers in the interval NyQiIiJJIm5HNiI=and if n is odd, then there are LCZJIm5HNiIjIiIiIiIjRiVGJg==

odd integers in the interval [1,n].NiI=For example, consider the following short table of examples.

(You can experiment with various values of n in A(n) by filling in a value in the entry box and then clicking on the button to evaluate the set function.)

Integer n

Maple function

Set of elements LUkqaW50ZXJzZWN0RyUqcHJvdGVjdGVkRzYkSSJBRzYiNyQiIiJJIm5HRic=

use DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # GetProperty( component_name, attribute_name ) # and # SetProperty( component_name, attribute_name, value ) # to access any of the attributes of the component. # See ?CustomizingComponents for more information. n := parse( GetProperty( 'TextArea0', 'value')); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LUkiQUc2IjYjIiNK

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

QyQtJkkmcGxvdHNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kqcG9pbnRwbG90R0YpNiktSSR6aXBHRig2JWYqNiRJInhHRilJInlHRilGKTYkSSlvcGVyYXRvckdGKUkmYXJyb3dHRilGKTckRjJGM0YpRilGKS1JKGNvbnZlcnRHRic2JC1JIkFHRik2IyIjXUklbGlzdEdGJzcjLUkkc2VxR0YnNiQiIiEvSSJpR0YpOyIiIi1JJW5vcHNHRic2I0Y7L0kmY29sb3JHRilJJHJlZEdGKS9JJ3N5bWJvbEdGJ0knY2lyY2xlR0YpL0kreHRpY2ttYXJrc0dGKSIiJi9JK3l0aWNrbWFya3NHRilGRC9JK3N5bWJvbHNpemVHRikiIzUvSSZ0aXRsZUdGKVEyVmlld35vZn50aGV+c2V0fkFGKUZI

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

A view of a portion of the set A to illustrate the density is 1/2 by noting that approximately half of the number line is covered.

Then the sequence of approximate densities of integers in A in the interval NyQiIiJJIm5HNiI=is (in the right-hand cells you can experiment with various values of n by changing the value of n in the entry box.)

KiZJKEludGVnZXJHJSpwcm90ZWN0ZWRHIiIiSSJuRzYiRiU=

Maple function

Sequence of approximate densities:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LUkkc2VxRyUqcHJvdGVjdGVkRzYkKiYtSSVjZWlsRzYiNiMsJComIyIiIiIiI0YuSSJuR0YpRi5GLkYuRjAhIiIvRjA7Ri4iI0k=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

NiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Another slightly more explicit way to express the fraction of integers in A in [1,n] is KiYtSSIrRyUqcHJvdGVjdGVkRzYkLCRJIm5HNiIjIiIiIiIjLCYjRisiIiVGKykhIiIsJkYoRitGK0YrRi5GK0YoRjE=. You can check it out with the following table, experimenting by by filling in a value in the entry box and then clicking on the button to evaluate the set function

Integer n

Maple command

Sequence of Approximate Densities

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

The limit of the sequence is

LUkmbGltaXRHNiI2JComLUkiK0clKnByb3RlY3RlZEc2JCwkSSJuR0YkIyIiIiIiIywmI0YuIiIlRi4pISIiLCZGLEYuRi5GLkYxRi5GLEY0L0YsSSlpbmZpbml0eUdGKQ==

print();

IyIiIiIiIw==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LSZJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJKnBvaW50cGxvdEdGKDYlNyMtSSRzZXFHRiY2JDckSSJuR0YoLUkuYXBwcm94RGVuc2l0eUdGKDYkSSJBR0YoRjEvRjE7IiIjIiRdIi9JJmNvbG9yR0YoSSRyZWRHRigvSSZ0aXRsZUdGKFE1RGVuc2l0eX5vZn50aGV+c2V0fkFGKA==

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

KipJJVBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIiIiJJI29mR0YnIiIjSSlzZXF1ZW5jZUdGJ0YoSSxhcHByb3hpbWF0ZUdGJ0Yo densities, showing how the limit of 1/2 is approached as n increases.

<Text-field style="Heading 3" italic="false" layout="Heading 3">The Set B and its density </Text-field>

Let B be the set of even integers in the range Oy1JJU9wZW5HNiI2IykiIiMsJkkibUdGJUYoISIjIiIiKUYoLCZGKkYoISIiRiw= 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together with the odd integers in the range

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You should experiment with various values of n in B(n) to get some feeling for B(n). (In the right-hand cells you can experiment with various values of n by changing the value of n in the entry box.)

Integer n

KiZJJk1hcGxlRzYiIiIiSShjb21tYW5kR0YkRiU=

LUkqaW50ZXJzZWN0RyUqcHJvdGVjdGVkRzYkKipJJFNldEc2IiIiIkkjb2ZHRihGKUkpZWxlbWVudHNHRihGKUkiQkdGKEYpNyRGKUkibkdGKA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

There are LyIiKSkiIiMtSSIrRyUqcHJvdGVjdGVkRzYkIiInISIk even integers in the interval 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there are LyIjOykiIiMtSSIrRyUqcHJvdGVjdGVkRzYkIiInISIjodd integers in the interval LzstSSVPcGVuRzYiNiMiI0siI2s7LUYlNiMpIiIjLUkiK0clKnByb3RlY3RlZEc2JCIiJyEiIkYp

More generally, there are LykiIiMsJkkibUc2IkYkISIkIiIiLCYpRiQsJkYmRiQhIiJGKSNGKUYkKUYkLCZGJkYkISIjRikjRi1GJA== even integers in the interval 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 and there are LykiIiMsJkkibUc2IkYkISIjIiIiLCYpRiQsJEYmRiQjRilGJClGJCwmRiZGJCEiIkYpI0YwRiQ=odd integers in the interval

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

QyQtJkkmcGxvdHNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kqcG9pbnRwbG90R0YpNiktSSR6aXBHRik2JWYqNiRJInhHRilJInlHRilGKTYkSSlvcGVyYXRvckdGKUkmYXJyb3dHRilGKTckRjJGM0YpRilGKS1JKGNvbnZlcnRHRic2JC1JIkJHRik2IyIjXUklbGlzdEdGJzcjLUkkc2VxR0YnNiQiIiEvSSJpR0YpOyIiIi1JJW5vcHNHRic2I0Y7L0kmY29sb3JHRilJJHJlZEdGKS9JJ3N5bWJvbEdGJ0knY2lyY2xlR0YpL0kreHRpY2ttYXJrc0dGKSIiJi9JK3l0aWNrbWFya3NHRilGRC9JK3N5bWJvbHNpemVHRikiIzUvSSZ0aXRsZUdGKVEyVmlld35vZn50aGV+c2V0fkJGKUZI

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

A view of a portion of the set B, to illustrate the density is 1/2 by noting that approximately half of the number line is covered.

Let n\342\211\2452 be a given integer, and let N = \342\214\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be the greatest integer less than LSZJJGxvZ0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjIiIjNiNJIm5HRig=so that 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where MzEiIiFJInBHNiIxRiUsJikiIiNJIk5HRiYiIiIkISIiRiRGLA==Case 1: N NiI=is 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 Then there are

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LCYqJiMiIiIiI0NGJSkiIiUsJkkibUc2IkYlRiVGJUYlRiUjIiIjIiIkISIi

LUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQpIiIjLCYqJkYqIiIiSSJqR0YnRi1GLSIiJCEiIi9GLjtGKkkibUdGJw==

LCYqJiMiIiIiI0NGJSkiIiUsJkkibUc2IkYlRiVGJUYlRiUjIiIjIiIkISIi

print();

LCYqJiMiIiIiI0NGJSkiIiUsJkkibUc2IkYlRiVGJUYlRiUjIiIjIiIkISIi

even integers in the ranges (4, 8], (16, 32], ... , 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

Subcase a: p is odd. Then n is in an interval of the form Kio2JCkiIiMsJkkibUc2IkYlISIiIiIiKUYlLCRGJ0YlRipJK2NvbnRhaW5pbmdHRihGKkkkb2RkRyUqcHJvdGVjdGVkR0YqSSlpbnRlZ2Vyc0dGKEYq. Then there are 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

Subcase b: p is even. Then n is in an interval of the form 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Then there are 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

Case 2 : N = LCRJIm1HNiIiIiM=is even.

Then there are

LUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQpIiIjLCYqJkYqIiIiSSJqR0YnRi1GLUYqISIiL0YuO0YtSSJtR0Yn

LCYqJiMiIiIiIzdGJSkiIiUsJkkibUc2IkYlRiVGJUYlRiUjRiUiIiQhIiI=

print();

LCYqJiMiIiIiIzdGJSkiIiUsJkkibUc2IkYlRiVGJUYlRiUjRiUiIiQhIiI=

odd integers in the ranges (2, 4], (8, 16], ... , 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

Subcase a: p is odd. Then n is in an interval of the form Kio2JCkiIiMsJEkibUc2IkYlKUYlLCZGJ0YlIiIiRitGK0krY29udGFpbmluZ0dGKEYrSSVldmVuRyUqcHJvdGVjdGVkR0YrSSlpbnRlZ2Vyc0dGKEYr. Then there are 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 Subcase b: p is even. Then n is in an interval of the form 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 there are 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

Summarizing the observations so far, there are

LUkiK0clKnByb3RlY3RlZEc2JiwkKiYjIiIiIiNDRikpIiIlLCZJIm1HNiJGKUYpRilGKUYpIyEiIyIiJCwkKiYjRikiIzdGKUYrRilGKSMhIiJGMg==

LCYqJiMiIiIiIilGJSkiIiUsJkkibUc2IkYlRiVGJUYlRiVGJSEiIg==

print();

LCYqJiMiIiIiIilGJSkiIiUsJkkibUc2IkYlRiVGJUYlRiVGJSEiIg==

NiI=integers in the set B in the interval 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

Case 1, subcase a: L0kibkc2IiwmKSIiI0kiTkdGJCIiIiomSSJwR0YkRilJJndoZXJlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkRilGKQ==L0kiTkc2IiwmSSJtR0YkIiIjISIiIiIiis odd and SSJwRzYiis odd. Then there are

LUkiK0clKnByb3RlY3RlZEc2JC1JKXNpbXBsaWZ5RzYkRiRJKF9zeXNsaWJHNiI2Iy1GIzYkLCYqJiMiIiIiI0NGMSkiIiUsJkkibUdGKkYxRjFGMUYxRjEjIiIjIiIkISIiLCYqJiNGMSIjN0YxKUY0RjZGMUYxI0YxRjlGOiwmKiYjRjFGOEYxSSJwR0YqRjFGMUZDRjE=

LCgpIiIjLCYqJkYkIiIiSSJtRzYiRidGJ0YkISIiRicjRidGJEYqKiZGK0YnSSJwR0YpRidGJw==

print();

LCgpIiIjLCYqJkYkIiIiSSJtRzYiRidGJ0YkISIiRicjRidGJEYqKiZGK0YnSSJwR0YpRidGJw==

SSlpbnRlZ2Vyc0c2Ig==in the set B in the range (1,n]. Notice that this can be rewritten as 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

Case 1, subcase b: My9JIm5HNiIsJikiIiNJIk5HRiUiIiIqKEkicEdGJUYqSSZ3aGVyZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYqRilGKkYqL0YmLCZJIm1HRiVGKCEiIkYqL0kibkc2IiwmKSIiI0kiTkdGJCIiIiooSSJwR0YkRilJJndoZXJlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkRilGKEYpRik=is odd and SSJwRzYiis even. Then there are

LCYtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLUkiK0dGJjYkLCYqJiMiIiIiI0NGMCkiIiUsJkkibUdGKEYwRjBGMEYwRjAjIiIjIiIkISIiLCYqJiNGMCIjN0YwKUYzRjVGMEYwI0YwRjhGOUYwKiYjRjBGN0YwSSJwR0YoRjBGMA==

LCgpIiIjLCYqJkYkIiIiSSJtRzYiRidGJ0YkISIiRidGJ0YqKiYjRidGJEYnSSJwR0YpRidGJw==

print();

LCgpIiIjLCYqJkYkIiIiSSJtRzYiRidGJ0YkISIiRidGJ0YqKiYjRidGJEYnSSJwR0YpRidGJw==

print();

SSlpbnRlZ2Vyc0c2Ig==in the set B in the range (1,n]. Notice that this can be rewritten as 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

Case 2, subcase a: My9JIm5HNiIsJikiIiNJIk5HRiUiIiIqKEkicEdGJUYqSSZ3aGVyZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYqRilGKkYqL0YmLCRJIm1HRiVGKA==is even and SSJwRzYiis odd. Then there are

LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywoKiYjIiIiIiIpRiwpIiIlLCZJIm1HRidGLEYsRixGLEYsIyIiJCIiIyEiIiomI0YsRjRGLEkicEdGJ0YsRiw=

LCgqJiMiIiIiIiNGJSkiIiVJIm1HNiJGJUYlIyIiJEYmISIiKiZGJEYlSSJwR0YqRiVGJQ==

print();

LCgqJiMiIiIiIiNGJSkiIiVJIm1HNiJGJUYlIyIiJEYmISIiKiZGJEYlSSJwR0YqRiVGJQ==

print();

SSlpbnRlZ2Vyc0c2Ig==in the set B in the range (1,n]. Notice that this can be rewritten as

LCYqJiMiIiIiIiNGJUkibkc2IkYlRiUjIiIkRiYhIiI=

print();

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiVGKy1GIzYlRistRiM2JUYrLUkjbW9HRiQ2MFEiLkYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQy8lKXN0cmV0Y2h5R0ZDLyUqc3ltbWV0cmljR0ZDLyUobGFyZ2VvcEdGQy8lLm1vdmFibGVsaW1pdHNHRkMvJSdhY2NlbnRHRkMvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZVLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnRitGK0YrRis=

Case 2, subcase b: L0kibkc2IiwmKSIiI0kiTkdGJCIiIiomSSJwR0YkRilJJndoZXJlRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkRilGKQ== L0kiTkc2IiwkSSJtR0YkIiIjis even and SSJwRzYiis even. Then there are

LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywoKiYjIiIiIiIpRiwpIiIlLCZJIm1HRidGLEYsRixGLEYsRiwhIiIqJiNGLCIiI0YsSSJwR0YnRixGLA==

LCgqJiMiIiIiIiNGJSkiIiVJIm1HNiJGJUYlRiUhIiIqJkYkRiVJInBHRipGJUYl

print();

LCgqJiMiIiIiIiNGJSkiIiVJIm1HNiJGJUYlRiUhIiIqJkYkRiVJInBHRipGJUYl

print();

SSlpbnRlZ2Vyc0c2Ig==in the set B in the range (1,n]. Notice that this can be rewritten as LCZJIm5HNiIjIiIiIiIjJCEiIiIiIUYmNiI=

<Text-field style="Heading 4" layout="Heading 4">Examples:</Text-field>

Example, Case 1, Subcase a: Take n = 15,

LUkmZmxvb3JHNiI2Iy1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLSZJJGxvZ0c2JEYoSShfc3lzbGliR0YkNiMiIiM2IyIjOg==

IiIk

print();

IiIk

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

so IiIp = 8, so 2m-1 = 3, so m=2 and p = 7, i.e.

LUkiK0clKnByb3RlY3RlZEc2JCIiKSIiKA==

IiM6

print();

IiM6

, then there are

LUkiK0clKnByb3RlY3RlZEc2JCMiIzoiIiMjISIiRig=

IiIo

print();

IiIo

integers in B(15), namely {3, 6, 8, 9,11,13,15}.

Example, Case 1, Subcase a: Take n=59,

LUkmZmxvb3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JJmV2YWxmR0YlNiMtJkkkbG9nR0YkNiMiIiM2IyIjZg==

IiIm

print();

IiIm

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

LUkiK0clKnByb3RlY3RlZEc2JCIjSyIjRg==

IiNm

print();

IiNm

and

there are

LUkiK0clKnByb3RlY3RlZEc2JCMiI2YiIiMjISIiRig=

IiNI

print();

IiNI

NiI=integers in the set B in the range [1,n], namely {3,6,8,9,11, 13,15,18,20,22, 24,26,28,30,32, 33,35,37,39,41, 43,45,47,49,51, 53,55,57,59}.

Example, Case 1, Subcase b: Take n =60 ,

LUkmZmxvb3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JJmV2YWxmR0YlNiMtJkkkbG9nR0YkNiMiIiM2IyIjZw==

IiIm

print();

IiIm

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

so IiNL = 32, so 2m-1 = 5, so m=3 and p = 28, i.e.

LUkiK0clKnByb3RlY3RlZEc2JCIjSyIjRw==

IiNn

print();

IiNn

, then there are

LUkiK0clKnByb3RlY3RlZEc2JCIjSSEiIg==

IiNI

print();

IiNI

integers in

LUkiQkc2IjYjIiNn

PD8iIiQiIiciIikiIioiIzYiIzgiIzoiIz0iIz8iI0EiI0MiI0UiI0ciI0kiI0siI0wiI04iI1AiI1IiI1QiI1YiI1giI1oiI1wiI14iI2AiI2IiI2QiI2Y=

print();

PD8iIiQiIiciIikiIioiIzYiIzgiIzoiIz0iIz8iI0EiI0MiI0UiI0ciI0kiI0siI0wiI04iI1AiI1IiI1QiI1YiI1giI1oiI1wiI14iI2AiI2IiI2QiI2Y=

Example, Case 2, Subcase a: Take n =29 ,

LUkmZmxvb3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JJmV2YWxmR0YlNiMtJkkkbG9nR0YkNiMiIiM2IyIjSA==

IiIl

print();

IiIl

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

so IiM7=16, so 2m = 4, so m=2 and p = 13, i.e.

LUkiK0clKnByb3RlY3RlZEc2JCIjOyIjOA==

IiNI

print();

IiNI

, then there are

LUkiK0clKnByb3RlY3RlZEc2JCMiI0giIiMjISIkRig=

IiM4

print();

IiM4

integers in

LUkiQkc2IjYjIiNI

PC8iIiQiIiciIikiIioiIzYiIzgiIzoiIz0iIz8iI0EiI0MiI0UiI0c=

print();

PC8iIiQiIiciIikiIioiIzYiIzgiIzoiIz0iIz8iI0EiI0MiI0UiI0c=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW9HRiQ2MFEiLkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZGLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYn

Example, Case 2, Subcase a: Take n = 17,

LUkmZmxvb3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JJmV2YWxmR0YlNiMtJkkkbG9nR0YkNiMiIiM2IyIjPA==

IiIl

print();

IiIl

and p = 1, i.e.

LUkiK0clKnByb3RlY3RlZEc2JCIjOyIiIg==

IiM8

print();

IiM8

, and there are

LUkiK0clKnByb3RlY3RlZEc2JCMiIzwiIiMjISIkRig=

IiIo

print();

IiIo

integers in B less than or equal to 17, namely {3, 6, 8, 9,11,13,15}.

Example, Case 2, Subcase b: Take n = 2,

LUkmZmxvb3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JJmV2YWxmR0YlNiMtJkkkbG9nR0YkNiMiIiNGLw==

IiIi

print();

IiIi

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

so IiIj=2, so m=2 and p = 0, i.e.

LUkiK0clKnByb3RlY3RlZEc2JCIiIyIiIQ==

IiIj

print();

IiIj

, then there are

LUkiK0clKnByb3RlY3RlZEc2JCIiIiEiIg==

IiIh

print();

IiIh

NiI=integers in B less than or equal to 2.

NiI=

The density of the set B is

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By the cases above, this limit will be squeezed between

LUkmbGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComLCYqJiMiIiIiIiNGLUkibkdGJ0YtRi0jIiIkRi4hIiJGLUYvRjIvRi9JKWluZmluaXR5R0Yl

IyIiIiIiIw==

print();

IyIiIiIiIw==

and

LUkmbGltaXRHNiI2JComLCYqJiMiIiIiIiNGKkkibkdGJEYqRipGKSEiIkYqRixGLS9GLEkpaW5maW5pdHlHJSpwcm90ZWN0ZWRH

IyIiIiIiIw==

print();

IyIiIiIiIw==

, so the density of B is 1/2.

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

LSZJJnBsb3RzRzYiNiNJKnBvaW50cGxvdEdGJTYlNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQ3JEkibkdGJS1JLmFwcHJveERlbnNpdHlHRiU2JEkiQkdGJUYvL0YvOyIiIyIkXSIvSSZjb2xvckdGJUkkcmVkR0YlL0kmdGl0bGVHRiVRNURlbnNpdHl+b2Z+dGhlfnNldH5CRiU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnKipJJVBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIiIiJJI29mR0YnIiIjSSlzZXF1ZW5jZUdGJ0YoSSxhcHByb3hpbWF0ZUdGJ0Yo densities, showing how the limit of 1/2 is approached as n increases.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

<Text-field style="Heading 3" italic="false" layout="Heading 3">The intersection of A and B and its density</Text-field>

A \342\213\202 B contains only odd integers in the ranges (2, 4], (8, 16], ... , 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

Then there are

LUkkc3VtRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQpIiIjLCYqJkYqIiIiSSJqR0YnRi1GLUYqISIiL0YuO0YtSSJtR0Yn

LCYqJiMiIiIiIzdGJSkiIiUsJkkibUc2IkYlRiVGJUYlRiUjRiUiIiQhIiI=

print();

LCYqJiMiIiIiIzdGJSkiIiUsJkkibUc2IkYlRiVGJUYlRiUjRiUiIiQhIiI=

Consider first the sequence of integers L0kibkc2IikiIiMsJEkibUdGJEYm, for 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 Then the number of integers in LUkqaW50ZXJzZWN0RyUqcHJvdGVjdGVkRzYkSSJBRzYiSSJCR0Yn in the interval [1,KSIiIywkSSJtRzYiRiM=] is 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Then the density of integers in the interval is

LUkmbGltaXRHNiI2JComLCYqJiMiIiIiIzdGKikiIiUsJkkibUdGJEYqRipGKkYqRiojRioiIiQhIiJGKikiIiMsJComRjRGKkYvRipGKkYyL0YvSSlpbmZpbml0eUclKnByb3RlY3RlZEc=

IyIiIiIiJA==

print();

IyIiIiIiJA==

Consider next the sequence of integers L0kibkc2IikiIiMsJkkibUdGJEYmIiIiRik=, for 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 Then the number of integers in LUkqaW50ZXJzZWN0RyUqcHJvdGVjdGVkRzYkSSJBRzYiSSJCR0Yn in the interval [1,KSIiIywmSSJtRzYiRiMiIiJGJw==] is still 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The density of integers in the interval is

LUkmbGltaXRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JComLCYqJiMiIiIiIzdGLSkiIiUsJkkibUdGJ0YtRi1GLUYtRi0jRi0iIiQhIiJGLSkiIiMsJiomRjdGLUYyRi1GLUYtRi1GNS9GMkkpaW5maW5pdHlHRiU=

IyIiIiIiJw==

print();

IyIiIiIiJw==

print();

Since the 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 then the limit does not exist and the intersection of the two sets does not have a well-defined density.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkNyMtSSRzZXFHRiU2JDckKSIiI0kiakdGJy1JJmV2YWxmR0YlNiMqJi1JJW5vcHNHRiU2Iy1JKmludGVyc2VjdEdGJTYkLUkiQUdGJzYjRi4tSSJCR0YnRj0iIiJGLiEiIi9GMDtGLyIjOi9JJnRpdGxlR0YnUTtEZW5zaXR5fmF0fnRoZX5pbmVnZXJzfjJeakYn

NiYtSSdDVVJWRVNHNiI2JDcwNyQkIiIlIiIhJCIzKysrKysrKytEISM9NyQkIiIpRiskIjMrKysrKysrXTdGLjckJCIjO0YrJCIzKysrKysrK0RKRi43JCQiI0tGKyQiMysrKysrK11pOkYuNyQkIiNrRiskIjMrKysrKytEIkckRi43JCQiJEciRiskIjMrKysrK11pUztGLjckJCIkYyNGKyQiMysrKysrREo/TEYuNyQkIiQ3JkYrJCIzKysrK11pOmc7Ri43JCQiJUM1RiskIjMrKysrRCJ5K0wkRi43JCQiJVs/RiskIjMnKioqKioqPjFSXW0iRi43JCQiJSc0JUYrJCIzKSoqKioqKjRgPkRMJEYuNyQkIiUjPilGKyQiMy4rKyttKGZpbSJGLjckJCImJVE7RiskIjMtKysrJCkpSEpMJEYuNyQkIiZvRiRGKyQiMygqKioqKio0JVxjbTtGLi1JJ0NPTE9VUkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYmSSRSR0JHRlxwJCIjNSEiIiRGK0YrRmRwLUkrQVhFU0xBQkVMU0dGXHA2JFEhRiVGaHAtSSZUSVRMRUdGXHA2I1E7RGVuc2l0eX5hdH50aGV+aW5lZ2Vyc34yXmpGJS1JJVZJRVdHRlxwNiRJKERFRkFVTFRHRiVGYHE=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LSZJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJKnBvaW50cGxvdEdGKDYlNyMtSSRzZXFHRiY2JDckSSJuR0YoLUkuYXBwcm94RGVuc2l0eUdGKDYkSSZBYW5kQkdGKEYxL0YxOyIiIyIkXSIvSSZjb2xvckdGKEkkcmVkR0YoL0kmdGl0bGVHRihRQURlbnNpdHl+b2Z+dGhlfnNldH5BfmludGVyc2VjdH5CRig=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnKipJJVBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiIiIiJJI29mR0YnIiIjSSlzZXF1ZW5jZUdGJ0YoSSxhcHByb3hpbWF0ZUdGJ0Yo densities of LUkqaW50ZXJzZWN0RyUqcHJvdGVjdGVkRzYkSSJBRzYiSSJCR0Yn, showing how the limit does not exist.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn