{"id":"https://openalex.org/W7225640834","doi":"https://doi.org/10.5281/zenodo.24085264","title":"The Divide-or-Multiply Sequence: Exact Structure and an Optimal Algorithm","display_name":"The Divide-or-Multiply Sequence: Exact Structure and an Optimal Algorithm","publication_year":2026,"publication_date":"2026-11-09","ids":{"openalex":"https://openalex.org/W7225640834","doi":"https://doi.org/10.5281/zenodo.24085264"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24085264","doi":"https://doi.org/10.5281/zenodo.24085264","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24085264","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","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24085264","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029786996","display_name":"Mark Kim","orcid":"https://orcid.org/0000-0001-7601-831X","observed_orcids":["https://orcid.org/0000-0001-7601-831X"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Kim, Mark","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0001-0181-5717","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5029786996"],"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":0.7250999808311462,"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":0.7250999808311462,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.20640000700950623,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T13806","display_name":"Mathematical and Computational Methods","score":0.030500000342726707,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/prime-exponents","display_name":"prime exponents","score":0.6589999794960022},{"id":"https://openalex.org/keywords/integer-factorization","display_name":"integer factorization","score":0.8799999952316284},{"id":"https://openalex.org/keywords/divisibility-conditions","display_name":"divisibility conditions","score":0.7210000157356262},{"id":"https://openalex.org/keywords/computational-lower-bounds","display_name":"computational lower bounds","score":0.6660000085830688},{"id":"https://openalex.org/keywords/optimal-algorithms","display_name":"optimal algorithms","score":0.35499998927116394}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7659000158309937},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.7630000114440918},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.6700999736785889},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.5631999969482422},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5065000057220459},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4982999861240387},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4900999963283539},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.44769999384880066},{"id":"https://openalex.org/C195292467","wikidata":"https://www.wikidata.org/wiki/Q2091879","display_name":"Log-log plot","level":3,"score":0.4399000108242035},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3982999920845032},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.34459999203681946},{"id":"https://openalex.org/C30860621","wikidata":"https://www.wikidata.org/wiki/Q49008","display_name":"Prime factor","level":3,"score":0.3370000123977661},{"id":"https://openalex.org/C9991821","wikidata":"https://www.wikidata.org/wiki/Q858520","display_name":"Divisibility rule","level":2,"score":0.3319999873638153},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.325300008058548},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.3237000107765198},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.3206999897956848},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.30250000953674316},{"id":"https://openalex.org/C162319229","wikidata":"https://www.wikidata.org/wiki/Q175263","display_name":"Data structure","level":2,"score":0.29510000348091125},{"id":"https://openalex.org/C71559656","wikidata":"https://www.wikidata.org/wiki/Q671298","display_name":"Divide and conquer algorithms","level":2,"score":0.2906000018119812},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.2897000014781952},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.2770000100135803},{"id":"https://openalex.org/C2779760435","wikidata":"https://www.wikidata.org/wiki/Q5396169","display_name":"Minor (academic)","level":2,"score":0.2547000050544739}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24085264","doi":"https://doi.org/10.5281/zenodo.24085264","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24085264","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.24085265","doi":"https://doi.org/10.5281/zenodo.24085265","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24085265","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.24085264","doi":"https://doi.org/10.5281/zenodo.24085264","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24085264","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W6944419975","https://openalex.org/W2511191713","https://openalex.org/W7214495143","https://openalex.org/W5531810","https://openalex.org/W2100346102","https://openalex.org/W2990049047","https://openalex.org/W2013230110","https://openalex.org/W6931334274","https://openalex.org/W1974322345","https://openalex.org/W7172282096"],"abstract_inverted_index":{"Let":[0],"a(1)=1":[1],"and":[2,8,43,149,205],"a(m+1)=a(m)/m":[3],"if":[4],"m":[5,23,40],"divides":[6,24,41],"a(m),":[7,42],"a(m+1)=m*a(m)":[9],"otherwise":[10],"(OEIS":[11],"A008336).":[12],"We":[13],"prove":[14],"the":[15,32,37,46,66,84,101,132,169,173,177,193],"product":[16],"identity":[17],"a(n)":[18,71,97,103,120,154],"=":[19],"(n-1)!":[20],"/":[21,91,125,157],"product_{m<n,":[22],"a(m)}":[25],"m^2,":[26],"establish":[27],"an":[28],"exact":[29,67],"recursion":[30,64],"on":[31],"prime":[33,50,68],"exponents":[34],"that":[35,45,118,131],"decides":[36],"divisibility":[38],"condition":[39],"show":[44],"exponent":[47,63,134,194],"of":[48,70,143,176,187],"a":[49,56,140],"p":[51],">":[52],"sqrt(n)":[53],"behaves":[54],"as":[55],"ballot-type":[57],"walk":[58],"with":[59],"fixed":[60],"parity.":[61],"The":[62],"yields":[65],"factorization":[69,175],"in":[72,121,155,181,203,210],"O(n":[73],"log":[74,75,92,126,146,147,158],"n)":[76,93,127,159],"arithmetic":[77],"operations":[78],"using":[79,184],"O(n)":[80],"memory,":[81],"completely":[82],"outperforming":[83],"traditional":[85],"big-integer":[86,201],"iteration":[87,202],"which":[88],"requires":[89,123],"Omega(n^2":[90],"word":[94],"operations.":[95],"Since":[96],">=":[98],"product_{n/2<p<n}":[99],"p,":[100],"number":[102],"has":[104],"at":[105],"least":[106],"(1/(2":[107],"ln":[108],"2)":[109],"+":[110],"o(1))":[111],"*":[112],"n":[113,145],"bits.":[114],"Therefore,":[115],"any":[116],"algorithm":[117,151],"outputs":[119],"binary":[122],"Omega(n":[124],"steps.":[128,160],"This":[129],"proves":[130],"proposed":[133],"framework":[135,195],"is":[136],"optimal":[137],"up":[138],"to":[139],"minor":[141],"factor":[142],"O(log":[144],"n),":[148],"no":[150],"can":[152],"evaluate":[153],"o(n":[156],"All":[161],"theoretical":[162],"claims":[163],"are":[164],"rigorously":[165],"proved.":[166],"In":[167],"C,":[168],"compact":[170],"implementation":[171],"computes":[172],"complete":[174],"monster":[178],"term":[179],"a(10^9)":[180],"55.9":[182],"seconds":[183],"2":[185],"GB":[186],"memory.":[188],"On":[189],"smaller":[190],"scales":[191],"(n=10^5),":[192],"runs":[196],"17":[197],"times":[198,208],"faster":[199,209],"than":[200],"Python":[204],"approximately":[206],"500":[207],"C.":[211]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-10T00:00:00"}