FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W2063697321> ?p ?o ?g. }

• W2063697321 endingPage "1913" @default.
• W2063697321 startingPage "1903" @default.
• W2063697321 abstract "In 1973, Erdős proved that a positive proportion of numbers are not of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma left-parenthesis n right-parenthesis minus n> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma (n)-n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the sum of the proper divisors of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove the analogous result where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma> <mml:semantics> <mml:mi>σ<!-- σ --></mml:mi> <mml:annotation encoding=application/x-tex>sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is replaced with the sum-of-unitary-divisors function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Superscript asterisk> <mml:semantics> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>sigma ^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (which sums divisors <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis d comma n slash d right-parenthesis equals 1> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>(d, n/d) = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>), thus solving a problem of te Riele from 1976. We also describe a fast algorithm for enumerating numbers not in the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma left-parenthesis n right-parenthesis minus n> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma (n)-n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Superscript asterisk Baseline left-parenthesis n right-parenthesis minus n> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma ^*(n)-n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n minus phi left-parenthesis n right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>φ<!-- φ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>n-varphi (n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi> <mml:semantics> <mml:mi>φ<!-- φ --></mml:mi> <mml:annotation encoding=application/x-tex>varphi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Euler’s function." @default.
• W2063697321 created "2016-06-24" @default.
• W2063697321 creator A5007445361 @default.
• W2063697321 creator A5065324760 @default.
• W2063697321 date "2013-10-29" @default.
• W2063697321 modified "2023-09-25" @default.
• W2063697321 title "Variant of a theorem of Erdős on the sum-of-proper-divisors function" @default.
• W2063697321 cites W1612880716 @default.
• W2063697321 cites W2004580743 @default.
• W2063697321 cites W2005537236 @default.
• W2063697321 cites W2010462498 @default.
• W2063697321 cites W2015796533 @default.
• W2063697321 cites W2057440978 @default.
• W2063697321 cites W2390833879 @default.
• W2063697321 cites W339666274 @default.
• W2063697321 cites W390714252 @default.
• W2063697321 cites W4233795935 @default.
• W2063697321 cites W935713918 @default.
• W2063697321 cites W981007808 @default.
• W2063697321 doi "https://doi.org/10.1090/s0025-5718-2013-02775-5" @default.
• W2063697321 hasPublicationYear "2013" @default.
• W2063697321 type Work @default.
• W2063697321 sameAs 2063697321 @default.
• W2063697321 citedByCount "5" @default.
• W2063697321 countsByYear W20636973212015 @default.
• W2063697321 countsByYear W20636973212016 @default.
• W2063697321 countsByYear W20636973212018 @default.
• W2063697321 countsByYear W20636973212021 @default.
• W2063697321 countsByYear W20636973212022 @default.
• W2063697321 crossrefType "journal-article" @default.
• W2063697321 hasAuthorship W2063697321A5007445361 @default.
• W2063697321 hasAuthorship W2063697321A5065324760 @default.
• W2063697321 hasBestOaLocation W20636973211 @default.
• W2063697321 hasConcept C11413529 @default.
• W2063697321 hasConcept C154945302 @default.
• W2063697321 hasConcept C18903297 @default.
• W2063697321 hasConcept C2776321320 @default.
• W2063697321 hasConcept C2777299769 @default.
• W2063697321 hasConcept C33923547 @default.
• W2063697321 hasConcept C41008148 @default.
• W2063697321 hasConcept C86803240 @default.
• W2063697321 hasConceptScore W2063697321C11413529 @default.
• W2063697321 hasConceptScore W2063697321C154945302 @default.
• W2063697321 hasConceptScore W2063697321C18903297 @default.
• W2063697321 hasConceptScore W2063697321C2776321320 @default.
• W2063697321 hasConceptScore W2063697321C2777299769 @default.
• W2063697321 hasConceptScore W2063697321C33923547 @default.
• W2063697321 hasConceptScore W2063697321C41008148 @default.
• W2063697321 hasConceptScore W2063697321C86803240 @default.
• W2063697321 hasIssue "288" @default.
• W2063697321 hasLocation W20636973211 @default.
• W2063697321 hasOpenAccess W2063697321 @default.
• W2063697321 hasPrimaryLocation W20636973211 @default.
• W2063697321 hasRelatedWork W151193258 @default.
• W2063697321 hasRelatedWork W1529400504 @default.
• W2063697321 hasRelatedWork W1607472309 @default.
• W2063697321 hasRelatedWork W1871911958 @default.
• W2063697321 hasRelatedWork W1892467659 @default.
• W2063697321 hasRelatedWork W2123357356 @default.
• W2063697321 hasRelatedWork W2349865494 @default.
• W2063697321 hasRelatedWork W2808586768 @default.
• W2063697321 hasRelatedWork W2998403542 @default.
• W2063697321 hasRelatedWork W3101673024 @default.
• W2063697321 hasVolume "83" @default.
• W2063697321 isParatext "false" @default.
• W2063697321 isRetracted "false" @default.
• W2063697321 magId "2063697321" @default.
• W2063697321 workType "article" @default.