what proportion of positive integers have exactly N pairs of consecutive integer factors?
$begingroup$
This is a followon to this question, which asked for the proportion of integers that have (any) consecutive integer factors.
Now, what proportion has exactly one pair? Since every odd number is = (2*n+1), we have to reject any integer K with factors 2, n, (2n+1). It looks as though there will be some ugly formula that starts with {2,n} and disallows (2n+1) ... and , when 2^j is a factor of n, disallows 2^(j+1) as a factor of K.
So, as a followon, I'll leave the general case as well: what proportion of positive integers has exactly N pairs of consecutive factors?
number-theory
asked Dec 14 '18 at 15:44
Carl WitthoftCarl Witthoft
32618
$endgroup$
add a comment |
$begingroup$
This is a followon to this question, which asked for the proportion of integers that have (any) consecutive integer factors.
Now, what proportion has exactly one pair? Since every odd number is = (2*n+1), we have to reject any integer K with factors 2, n, (2n+1). It looks as though there will be some ugly formula that starts with {2,n} and disallows (2n+1) ... and , when 2^j is a factor of n, disallows 2^(j+1) as a factor of K.
So, as a followon, I'll leave the general case as well: what proportion of positive integers has exactly N pairs of consecutive factors?
number-theory
asked Dec 14 '18 at 15:44
Carl WitthoftCarl Witthoft
32618
$endgroup$
$begingroup$
Clearly deserves a vote.
$endgroup$
– marty cohen
Dec 14 '18 at 17:26
add a comment |
$begingroup$
This is a followon to this question, which asked for the proportion of integers that have (any) consecutive integer factors.
Now, what proportion has exactly one pair? Since every odd number is = (2*n+1), we have to reject any integer K with factors 2, n, (2n+1). It looks as though there will be some ugly formula that starts with {2,n} and disallows (2n+1) ... and , when 2^j is a factor of n, disallows 2^(j+1) as a factor of K.
So, as a followon, I'll leave the general case as well: what proportion of positive integers has exactly N pairs of consecutive factors?
number-theory
asked Dec 14 '18 at 15:44
Carl WitthoftCarl Witthoft
32618
$endgroup$
This is a followon to this question, which asked for the proportion of integers that have (any) consecutive integer factors.
Now, what proportion has exactly one pair? Since every odd number is = (2*n+1), we have to reject any integer K with factors 2, n, (2n+1). It looks as though there will be some ugly formula that starts with {2,n} and disallows (2n+1) ... and , when 2^j is a factor of n, disallows 2^(j+1) as a factor of K.
So, as a followon, I'll leave the general case as well: what proportion of positive integers has exactly N pairs of consecutive factors?
number-theory
number-theory
asked Dec 14 '18 at 15:44
Carl WitthoftCarl Witthoft
32618
asked Dec 14 '18 at 15:44
Carl WitthoftCarl Witthoft
32618
asked Dec 14 '18 at 15:44
Carl WitthoftCarl Witthoft
32618
asked Dec 14 '18 at 15:44
Carl WitthoftCarl Witthoft
32618
asked Dec 14 '18 at 15:44
Carl WitthoftCarl Witthoft
32618
32618
$begingroup$
Clearly deserves a vote.
$endgroup$
– marty cohen
Dec 14 '18 at 17:26
add a comment |
$begingroup$
Clearly deserves a vote.
$endgroup$
– marty cohen
Dec 14 '18 at 17:26
$begingroup$
Clearly deserves a vote.
$endgroup$
– marty cohen
Dec 14 '18 at 17:26
$begingroup$
Clearly deserves a vote.
$endgroup$
– marty cohen
Dec 14 '18 at 17:26
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039523%2fwhat-proportion-of-positive-integers-have-exactly-n-pairs-of-consecutive-integer%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039523%2fwhat-proportion-of-positive-integers-have-exactly-n-pairs-of-consecutive-integer%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
Clearly deserves a vote.
$endgroup$
– marty cohen
Dec 14 '18 at 17:26