{"id":"https://openalex.org/W7225040941","doi":"https://doi.org/10.5281/zenodo.23865007","title":"Long gaps, twin pairs and forbidden differences in the Ulam sequence","display_name":"Long gaps, twin pairs and forbidden differences in the Ulam sequence","publication_year":2026,"publication_date":"2026-11-01","ids":{"openalex":"https://openalex.org/W7225040941","doi":"https://doi.org/10.5281/zenodo.23865007"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23865007","doi":"https://doi.org/10.5281/zenodo.23865007","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23865007","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23865007","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151699530","display_name":"Shunsuke Funayama","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Funayama, Shunsuke","raw_affiliation_strings":["Independent researcher"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Independent researcher","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5151699530"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9916999936103821,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11166","display_name":"Analytic Number Theory Research","score":0.9916999936103821,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.003800000064074993,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12516","display_name":"Advanced Mathematical Theories and Applications","score":0.0010000000474974513,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/twin-pairs","display_name":"twin pairs","score":0.843999981880188},{"id":"https://openalex.org/keywords/middle-third","display_name":"middle third","score":0.4650000035762787},{"id":"https://openalex.org/keywords/outliers","display_name":"outliers","score":0.5249999761581421}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.79339998960495},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6732000112533569},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.61080002784729},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.5735999941825867},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.41659998893737793},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.36719998717308044},{"id":"https://openalex.org/C79337645","wikidata":"https://www.wikidata.org/wiki/Q779824","display_name":"Outlier","level":2,"score":0.32820001244544983},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.3179999887943268},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.30070000886917114},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.27709999680519104},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.27160000801086426},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.26260000467300415}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23865007","doi":"https://doi.org/10.5281/zenodo.23865007","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23865007","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.23865008","doi":"https://doi.org/10.5281/zenodo.23865008","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23865008","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.23865007","doi":"https://doi.org/10.5281/zenodo.23865007","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23865007","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4414019402","https://openalex.org/W7225041887","https://openalex.org/W4408237508","https://openalex.org/W7225056604","https://openalex.org/W1078482883","https://openalex.org/W7225038541","https://openalex.org/W2884714550","https://openalex.org/W1680596605","https://openalex.org/W2093556223","https://openalex.org/W4413456568"],"abstract_inverted_index":{"Erd\u0151s":[0],"problem":[1],"#342":[2],"asks":[3],"three":[4],"questions":[5],"about":[6,122,362,377,410],"the":[7,38,44,74,108,148,154,176,199,232,245,249,259,269,276,288,305,316,320,327,357,385,388,399,411,424,432,445,448,451,457,460],"Ulam":[8,412],"sequence":[9,413],"1,":[10],"2,":[11,65],"3,":[12,66],"4,":[13],"6,":[14,186,339],"8,":[15],"11,":[16],"\u2026;":[17],"one":[18,239],"of":[19,32,40,73,102,110,134,151,167,198,218,228,234,248,257,268,275,304,343,345,351,370,376,450],"them":[20],"is":[21,52,87,99,119,215,408,438],"whether":[22],"there":[23,209],"are":[24,292,418,428],"infinitely":[25,138],"many":[26,139],"pairs":[27,111,140],"a,":[28,112],"a":[29,79,91,100,113,144,216,224,235,254,374],"+":[30,114,127,213,334],"2":[31,115,214],"terms":[33,42,62,197,267,303],"(twin":[34],"pairs).":[35],"We":[36,55],"study":[37],"differences":[39,155,264,301],"consecutive":[41],"through":[43],"angles":[45],"{n\u03b2},":[46],"\u03b2":[47],"=":[48],"\u03b1/2\u03c0,":[49],"where":[50],"\u03b1":[51],"Steinerberger's":[53],"frequency.":[54],"assume":[56],"an":[57,403],"explicit":[58],"hypothesis":[59],"(H):":[60],"all":[61,272,299],"other":[63],"than":[64],"47,":[67],"69,":[68],"102":[69],"lie":[70,325],"within":[71],"26.6\u00b0":[72],"middle":[75,200,270,306],"third.":[76,201],"Under":[77,178,202],"(H),":[78,179,203,258],"term":[80,92],"whose":[81],"angle":[82,164,405],"lies":[83,364],"in":[84,356,421],"[213.3\u00b0,":[85],"240\u00b0)":[86],"always":[88],"followed":[89],"by":[90],"two":[93,103,135,196,219,266,289,394],"steps":[94,372],"later,":[95],"unless":[96],"that":[97,107,131,156,281],"number":[98,109,149],"sum":[101,217],"outliers.":[104,136,152,220],"It":[105],"follows":[106],"up":[116,285,360],"to":[117,171,286,297,361,379,441],"N":[118],"at":[120,353,447],"least":[121,354],"|U":[123],"\u2229":[124],"[1,":[125],"N]|/(52(|E_N|":[126],"1)),":[128],"minus":[129],"corrections":[130,396],"count":[132],"sums":[133],"So":[137],"would":[141],"follow":[142],"from":[143,231,341,384,402],"growth":[145],"condition":[146],"on":[147,326,365],"|E_N|":[150],"For":[153],"OEIS":[157],"A072540":[158],"lists":[159],"as":[160,263],"not":[161,283],"occurring,":[162],"short":[163],"arguments,":[165],"some":[166],"which":[168],"go":[169],"back":[170],"Gibbs":[172],"and":[173,181,205,253,294,347,387,393,423,459],"McCranie,":[174],"give":[175],"following.":[177],"11":[180],"33":[182],"never":[183],"occur.":[184],"Unconditionally,":[185],"16,":[187],"18,":[188],"21,":[189],"23,":[190],"26,":[191],"28":[192],"cannot":[193],"occur":[194,208,284],"between":[195,265,302],"14":[204],"31":[206],"can":[207],"only":[210,238],"when":[211],"n":[212],"With":[221,244],"(H)":[222],"alone,":[223],"finite":[225,425],"case":[226,426],"analysis":[227,261],"everything":[229],"visible":[230],"start":[233],"gap":[236,350],"adds":[237],"further":[240],"value":[241],"below":[242,251,279],"1500.":[243],"exact":[246],"list":[247,359],"outliers":[250,386],"10^4":[252],"4.7\u00b0":[255],"version":[256],"same":[260],"excludes,":[262],"third,":[271],"but":[273,435],"17":[274],"173":[277],"values":[278],"300":[280],"do":[282],"2^31;":[287],"smallest":[290],"survivors":[291],"54":[293],"102.":[295],"Up":[296],"2^31,":[298],"1.6\u00b710^8":[300],"third":[307],"agree":[308],"with":[309,336,431],"both":[310],"analyses.":[311],"Without":[312],"hypotheses":[313],"we":[314],"describe":[315],"long":[317],"gaps.":[318],"In":[319],"computed":[321],"data":[322],"their":[323,436],"lengths":[324],"lattice":[328],"521":[329],"or":[330],"530":[331],"plus":[332],"22a":[333],"259b":[335],"|a|":[337],"\u2264":[338],"built":[340,383],"denominators":[342],"convergents":[344],"\u03b2,":[346],"every":[348],"record":[349,358],"length":[352],"262":[355],"10^12":[363],"it.":[366],"One":[367],"more":[368],"turn":[369],"259":[371],"costs":[373],"factor":[375],"1/40":[378],"1/49.":[380],"A":[381],"formula":[382],"angular":[389],"distribution":[390],"gives":[391],"1/36,":[392],"measured":[395],"account":[397],"for":[398],"rest.":[400],"Apart":[401],"elementary":[404],"statement,":[406],"nothing":[407],"proved":[409],"without":[414],"hypotheses.":[415],"The":[416,453],"arguments":[417],"written":[419],"out":[420,430],"full":[422],"analyses":[427],"carried":[429],"deposited":[433],"code,":[434],"validity":[437],"left":[439],"open":[440],"external":[442],"verification;":[443],"see":[444],"note":[446],"end":[449],"paper.":[452],"accompanying":[454],"archive":[455],"contains":[456],"programs":[458],"recorded":[461],"outputs.":[462]},"counts_by_year":[],"updated_date":"2026-11-03T10:50:19.383937","created_date":"2026-11-02T00:00:00"}
