2b:null 2c:["$","title",null,{"children":"Prime factorization notations | Nat Alison | Observable"}] 31:T5fb,{ let width = 20; let height = 50; return html`

${_.range(1, drawLimit).map((n) => { let factors = [...allPrimeFactors[n]]; factors.reverse(); return html`

${factors.map((f) => { let color = getColor(f, Math.log2(drawLimit)); let opacity = 1; if (highlightPrimes && factors.length !== 1) { let hsl = d3.hsl(color); hsl.s *= 0.25; color = d3.color(hsl).toString(); opacity = 0.25; } else if (selectedPrime && f !== +selectedPrime) { let hsl = d3.hsl(color); hsl.s *= 0.25; color = d3.color(hsl).toString(); opacity = 0.25; } return html`

${f}

`; })}

`; }32:[] 2f:{"id":"a8a1bc97bded21aa","slug":null,"title":"Prime factorization notations","thumbnail":"0da636bebdd7a8c868494e0261d2a6422002420fb82ba68cf244aee2ffafcfbf","access_level":"public","create_time":"$D2024-08-25T22:57:57.215Z","update_time":"$D2024-11-09T09:08:40.633Z","view_time":null,"first_public_time":"$D2024-08-25T22:58:11.290Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"b2691018f1bbc657","login":"tesseralis","name":"Nat Alison","avatar_url":"https://avatars-next.observableusercontent.com/f0d364753a3a46bad109195a0a2beaa6160fb19714aeb0a1979f2d9b8beb1557","tier":"starter_2024"},"version":809,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"854148fc2192f6e1","login":"tesseralis","name":"Nat Alison","avatar_url":"https://avatars-next.observableusercontent.com/f0d364753a3a46bad109195a0a2beaa6160fb19714aeb0a1979f2d9b8beb1557","tier":null}],"members":[{"type":"individual","id":"854148fc2192f6e1","login":"tesseralis","name":"Nat Alison","avatar_url":"https://avatars-next.observableusercontent.com/f0d364753a3a46bad109195a0a2beaa6160fb19714aeb0a1979f2d9b8beb1557","tier":null,"role":"editor"},{"type":"team","id":"b2691018f1bbc657","login":"tesseralis","name":"Nat Alison","avatar_url":"https://avatars-next.observableusercontent.com/f0d364753a3a46bad109195a0a2beaa6160fb19714aeb0a1979f2d9b8beb1557","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":{"id":"aa587307bf066d1f","slug":"math","title":"Math!","owner":{"id":"b2691018f1bbc657","login":"tesseralis","name":"Nat Alison"}},"type":"notebook","body":{"title":"Prime factorization notations","cells":[{"id":0,"value":"# Prime factorization notations","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":116,"value":"## Primes as logarithmic lengths\n\nBy assigning each prime ${tex`p`} a length ${tex`\\log p`} we can represent each number as a _sum_ of its prime factors rather than a product.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":204,"value":"viewof highlightPrimes = Inputs.toggle({ label: \"Highlight Prime\" })","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":235,"value":"viewof selectedPrime = Inputs.select(\n [\"\", ..._.takeWhile(primes, (p) => p < drawLimit)],\n { label: \"Highlight Prime\" }\n)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":121,"value":"$31","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":802,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":759,"value":"function getColor(n, max) {\n let theta = frac(Math.log2(n));\n let shade = 1 - Math.log2(n) / (max);\n return `lch(${100 * shade}% 100 ${-25 + theta * 360})`;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":761,"value":"function frac(x) {\n return x - Math.floor(x);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":166,"value":"scale = {\n let maxPrime = primes.find((p) => p > drawLimit);\n return (p) => {\n return d3.interpolateSpectral(Math.log2(p) / Math.log2(maxPrime));\n };\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":163,"value":"drawLimit = 2 ** 10","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":147,"value":"scheme = [\n ...d3.schemeCategory10,\n ...d3.schemeTableau10,\n ...d3.schemeObservable10,\n ...d3.schemeSet1,\n ...d3.schemeSet2,\n ...d3.schemeSet3\n]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":140,"value":"primeIndexes = {\n let indexes = {};\n for (let i = 0; i < primes.length; i++) {\n indexes[primes[i]] = i;\n }\n return indexes;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":425,"value":"_.range(0, 17).map((n) => n / 16)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":377,"value":"primes.filter((p) => p < 32).map((p) => ({ p, log: 2 * Math.log2(p) }))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":255,"value":"## Primes by Gaps\n\nComparing primes by previous and next gaps gives the strong, weak, and balanced primes.\n\nCan we say anything about ${tex`p - 1`} for these classes?","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":257,"value":"primes.filter((p, i) => p - primes[i - 1] === primes[i + 1] - p)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":49,"value":"## Primes as smooth sums\n\nIf we write a prime ${tex`p = a + b`}, what is the minimal largest prime factor of ${tex`ab`}?\n\n- Record breakers seem to be [A000229](https://oeis.org/A000229) (something to do with quadratic residues) but I don't see the correlation.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":43,"value":"Plot.plot({\n marks: [Plot.dot(primeSums, { x: \"p\", y: \"smallest\", tip: true })]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"// Plot.plot({\n// marks: [Plot.dot(numberSums, { x: \"n\", y: \"smallest\", tip: true })]\n// })","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":30,"value":"primeSums = _.take(primes, 2000).map((n) => ({\n p: n,\n smallest: smallestSmoothnessSum(n)\n}))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":54,"value":"// numberSums = _.range(1000).map((n) => ({\n// n,\n// smallest: smallestSmoothnessSum(n)\n// }))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":66,"value":"// _.range(1000)\n// .map((n) => {\n// let smallest = numberSums[n].smallest;\n// let factors = _.uniq(allPrimeFactors[n]);\n// let primeSmallest = factors.map((p) => numberSums[p].smallest);\n// return {\n// n,\n// smallest,\n// primeSmallest\n// };\n// })\n// .filter((data) => data.primeSmallest.every((p) => p > data.smallest))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":24,"value":"function smallestSmoothnessSum(n) {\n let largest = Infinity;\n for (let i = 1; i < n / 2; i++) {\n let j = n - i;\n let iFactors = _.uniq(allPrimeFactors[i]);\n let jFactors = _.uniq(allPrimeFactors[j]);\n let iLargest = _.max(_.union(iFactors, jFactors));\n largest = Math.min(largest, iLargest);\n }\n return largest;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":76,"value":"## Binary Notation","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":85,"value":"Plot.plot({\n axis: null,\n margin: 20,\n marginRight: 120,\n height: 1500,\n marks: [\n Plot.tree(binaryPrimes, {\n path: \"path\",\n text: (d) => d?.p,\n fill: (d) => (d ? \"red\" : \"black\"),\n stroke: \"black\"\n })\n ]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":83,"value":"binaryPrimes = _.takeWhile(primes.slice(1), (p) => p < 2 ** 10).map((p) => ({\n path: digits(p, 2).join(\"/\"),\n p\n}))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":78,"value":"function digits(n, radix) {\n let digits = [];\n while (n !== 0) {\n digits.push(n % radix);\n n = Math.floor(n / radix);\n }\n return digits;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":14,"value":"allPrimeFactors = primeFactors(limit)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":18,"value":"limit = 2 ** 16","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"import { primes, primeFactors } from \"@tesseralis/math\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 30:"$2f:body"