On Thursday 29 November 2007, Eric Angelini wrote:
Start the sequence S with a(1)=1. The next term is the smallest one not yet present in S with the rule: the a(n) terms immediately after a(n) must be < a(n) or > 2*a(n). ... S starts like this (I think):
s = 1, 3, 2, 7, 15, 4, 31, 63, 127, 255, 5, 8, ...
The last term is wrong, because it's neither <5 nor >10. The sequence begins as follows: 1 2^2-1 2 2^3-1 2^4-1 4 2^5-1 2^6-1 2^7-1 2^8-1 5 2^9-1 2^10-1 2^11-1 2^12-1 2^13-1 6 2^14-1 2^15-1 2^16-1 2^17-1 2^18-1 2^19-1 8 2^20-1 2^21-1 2^22-1 2^23-1 2^24-1 2^25-1 2^26-1 2^27-1 9 2^28-1 2^29-1 2^30-1 2^31-1 2^32-1 2^33-1 2^34-1 2^35-1 2^36-1 10 2^37-1 2^38-1 2^39-1 2^40-1 2^41-1 2^42-1 2^43-1 2^44-1 2^45-1 2^46-1 11 2^47-1 2^48-1 2^49-1 2^50-1 2^51-1 2^52-1 2^53-1 2^54-1 2^55-1 2^56-1 2^57-1 12 2^58-1 2^59-1 2^60-1 2^61-1 2^62-1 2^63-1 2^64-1 2^65-1 2^66-1 2^67-1 2^68-1 2^69-1 13 2^70-1 2^71-1 2^72-1 2^73-1 2^74-1 2^75-1 2^76-1 2^77-1 2^78-1 2^79-1 2^80-1 2^81-1 2^82-1 14 2^83-1 2^84-1 2^85-1 2^86-1 2^87-1 2^88-1 2^89-1 2^90-1 2^91-1 2^92-1 2^93-1 2^94-1 2^95-1 2^96-1 16 2^97-1 2^98-1 2^99-1 2^100-1 2^101-1 2^102-1 2^103-1 2^104-1 2^105-1 2^106-1 2^107-1 2^108-1 2^109-1 2^110-1 2^111-1 2^112-1 17 2^113-1 2^114-1 2^115-1 2^116-1 2^117-1 2^118-1 2^119-1 2^120-1 2^121-1 2^122-1 2^123-1 2^124-1 2^125-1 2^126-1 2^127-1 2^128-1 2^129-1 18 2^130-1 2^131-1 2^132-1 2^133-1 2^134-1 2^135-1 2^136-1 2^137-1 2^138-1 2^139-1 2^140-1 2^141-1 2^142-1 2^143-1 2^144-1 2^145-1 2^146-1 2^147-1 19 2^148-1 2^149-1 2^150-1 2^151-1 2^152-1 2^153-1 2^154-1 2^155-1 2^156-1 2^157-1 2^158-1 2^159-1 2^160-1 2^161-1 2^162-1 2^163-1 2^164-1 2^165-1 2^166-1 20 2^167-1 2^168-1 2^169-1 2^170-1 2^171-1 2^172-1 2^173-1 2^174-1 2^175-1 2^176-1 2^177-1 2^178-1 2^179-1 2^180-1 2^181-1 2^182-1 2^183-1 2^184-1 2^185-1 2^186-1 21 2^187-1 2^188-1 2^189-1 2^190-1 2^191-1 2^192-1 2^193-1 2^194-1 2^195-1 2^196-1 2^197-1 2^198-1 2^199-1 2^200-1 2^201-1 2^202-1 2^203-1 2^204-1 2^205-1 2^206-1 2^207-1 22 2^208-1 2^209-1 2^210-1 2^211-1 2^212-1 2^213-1 2^214-1 2^215-1 2^216-1 2^217-1 2^218-1 2^219-1 2^220-1 2^221-1 2^222-1 2^223-1 2^224-1 2^225-1 2^226-1 2^227-1 2^228-1 2^229-1 23 2^230-1 2^231-1 2^232-1 2^233-1 2^234-1 2^235-1 2^236-1 2^237-1 2^238-1 2^239-1 2^240-1 2^241-1 2^242-1 2^243-1 2^244-1 2^245-1 2^246-1 2^247-1 2^248-1 2^249-1 2^250-1 2^251-1 2^252-1 24 2^253-1 2^254-1 2^255-1 2^256-1 2^257-1 2^258-1 2^259-1 2^260-1 2^261-1 2^262-1 2^263-1 2^264-1 2^265-1 2^266-1 2^267-1 2^268-1 2^269-1 2^270-1 2^271-1 2^272-1 2^273-1 2^274-1 2^275-1 2^276-1 25 2^277-1 2^278-1 2^279-1 2^280-1 2^281-1 2^282-1 2^283-1 2^284-1 2^285-1 2^286-1 2^287-1 2^288-1 2^289-1 2^290-1 2^291-1 2^292-1 2^293-1 2^294-1 2^295-1 2^296-1 2^297-1 2^298-1 2^299-1 2^300-1 2^301-1 26 2^302-1 2^303-1 2^304-1 2^305-1 2^306-1 2^307-1 2^308-1 2^309-1 2^310-1 2^311-1 2^312-1 2^313-1 2^314-1 2^315-1 2^316-1 2^317-1 2^318-1 2^319-1 2^320-1 2^321-1 2^322-1 2^323-1 2^324-1 2^325-1 2^326-1 2^327-1 27 2^328-1 2^329-1 2^330-1 2^331-1 2^332-1 2^333-1 2^334-1 2^335-1 2^336-1 2^337-1 2^338-1 2^339-1 2^340-1 2^341-1 2^342-1 2^343-1 2^344-1 2^345-1 2^346-1 2^347-1 2^348-1 2^349-1 2^350-1 2^351-1 2^352-1 2^353-1 2^354-1 28 2^355-1 2^356-1 2^357-1 2^358-1 2^359-1 2^360-1 2^361-1 2^362-1 2^363-1 2^364-1 2^365-1 2^366-1 2^367-1 2^368-1 2^369-1 2^370-1 2^371-1 2^372-1 2^373-1 2^374-1 2^375-1 2^376-1 2^377-1 2^378-1 2^379-1 2^380-1 2^381-1 2^382-1 29 2^383-1 2^384-1 2^385-1 2^386-1 2^387-1 2^388-1 2^389-1 2^390-1 2^391-1 2^392-1 2^393-1 2^394-1 2^395-1 2^396-1 2^397-1 2^398-1 2^399-1 2^400-1 2^401-1 2^402-1 2^403-1 2^404-1 2^405-1 2^406-1 2^407-1 2^408-1 2^409-1 2^410-1 2^411-1 30 2^412-1 2^413-1 2^414-1 2^415-1 2^416-1 2^417-1 2^418-1 2^419-1 2^420-1 2^421-1 2^422-1 2^423-1 2^424-1 2^425-1 2^426-1 2^427-1 2^428-1 2^429-1 2^430-1 2^431-1 2^432-1 2^433-1 2^434-1 2^435-1 2^436-1 2^437-1 2^438-1 2^439-1 2^440-1 2^441-1 32 2^442-1 2^443-1 2^444-1 2^445-1 2^446-1 2^447-1 2^448-1 2^449-1 2^450-1 2^451-1 2^452-1 2^453-1 2^454-1 2^455-1 2^456-1 2^457-1 2^458-1 2^459-1 2^460-1 2^461-1 2^462-1 2^463-1 2^464-1 2^465-1 2^466-1 2^467-1 2^468-1 2^469-1 2^470-1 2^471-1 2^472-1 2^473-1 33 2^474-1 2^475-1 2^476-1 2^477-1 2^478-1 2^479-1 2^480-1 2^481-1 2^482-1 2^483-1 2^484-1 2^485-1 2^486-1 2^487-1 2^488-1 2^489-1 2^490-1 2^491-1 2^492-1 2^493-1 2^494-1 2^495-1 2^496-1 2^497-1 2^498-1 2^499-1 2^500-1 2^501-1 2^502-1 2^503-1 2^504-1 2^505-1 2^506-1 34 2^507-1 2^508-1 2^509-1 2^510-1 2^511-1 2^512-1 2^513-1 2^514-1 2^515-1 2^516-1 2^517-1 2^518-1 2^519-1 2^520-1 2^521-1 2^522-1 2^523-1 2^524-1 2^525-1 2^526-1 2^527-1 2^528-1 2^529-1 2^530-1 2^531-1 2^532-1 2^533-1 2^534-1 2^535-1 2^536-1 2^537-1 2^538-1 2^539-1 2^540-1 35 2^541-1 2^542-1 2^543-1 2^544-1 2^545-1 2^546-1 2^547-1 2^548-1 2^549-1 2^550-1 2^551-1 2^552-1 2^553-1 2^554-1 2^555-1 2^556-1 2^557-1 2^558-1 2^559-1 2^560-1 2^561-1 2^562-1 2^563-1 2^564-1 2^565-1 2^566-1 2^567-1 2^568-1 2^569-1 2^570-1 2^571-1 2^572-1 2^573-1 2^574-1 2^575-1 36 2^576-1 2^577-1 2^578-1 2^579-1 2^580-1 2^581-1 2^582-1 2^583-1 2^584-1 2^585-1 2^586-1 2^587-1 2^588-1 2^589-1 2^590-1 2^591-1 2^592-1 2^593-1 2^594-1 2^595-1 2^596-1 2^597-1 2^598-1 2^599-1 2^600-1 2^601-1 2^602-1 2^603-1 2^604-1 2^605-1 2^606-1 2^607-1 2^608-1 2^609-1 2^610-1 2^611-1 37 2^612-1 2^613-1 2^614-1 2^615-1 2^616-1 2^617-1 2^618-1 2^619-1 2^620-1 2^621-1 2^622-1 2^623-1 2^624-1 2^625-1 2^626-1 2^627-1 2^628-1 2^629-1 2^630-1 2^631-1 2^632-1 2^633-1 2^634-1 2^635-1 2^636-1 2^637-1 2^638-1 2^639-1 2^640-1 2^641-1 2^642-1 2^643-1 2^644-1 2^645-1 2^646-1 2^647-1 2^648-1 38 2^649-1 2^650-1 2^651-1 2^652-1 2^653-1 2^654-1 2^655-1 2^656-1 2^657-1 2^658-1 2^659-1 2^660-1 2^661-1 2^662-1 2^663-1 2^664-1 2^665-1 2^666-1 2^667-1 2^668-1 2^669-1 2^670-1 2^671-1 2^672-1 2^673-1 2^674-1 2^675-1 2^676-1 2^677-1 2^678-1 2^679-1 2^680-1 2^681-1 2^682-1 2^683-1 2^684-1 2^685-1 2^686-1 39 2^687-1 2^688-1 2^689-1 2^690-1 2^691-1 2^692-1 2^693-1 2^694-1 2^695-1 2^696-1 2^697-1 2^698-1 2^699-1 2^700-1 2^701-1 2^702-1 2^703-1 2^704-1 2^705-1 2^706-1 2^707-1 2^708-1 2^709-1 2^710-1 2^711-1 2^712-1 2^713-1 2^714-1 2^715-1 2^716-1 2^717-1 2^718-1 2^719-1 2^720-1 2^721-1 2^722-1 2^723-1 2^724-1 2^725-1 40 2^726-1 2^727-1 2^728-1 2^729-1 2^730-1 2^731-1 2^732-1 2^733-1 2^734-1 2^735-1 2^736-1 2^737-1 2^738-1 2^739-1 2^740-1 2^741-1 2^742-1 2^743-1 2^744-1 2^745-1 2^746-1 2^747-1 2^748-1 2^749-1 2^750-1 2^751-1 2^752-1 2^753-1 2^754-1 2^755-1 2^756-1 2^757-1 2^758-1 2^759-1 2^760-1 2^761-1 2^762-1 2^763-1 2^764-1 2^765-1 41 2^766-1 2^767-1 2^768-1 2^769-1 2^770-1 2^771-1 2^772-1 2^773-1 2^774-1 2^775-1 2^776-1 2^777-1 2^778-1 2^779-1 2^780-1 2^781-1 2^782-1 2^783-1 2^784-1 2^785-1 2^786-1 2^787-1 2^788-1 2^789-1 2^790-1 2^791-1 2^792-1 2^793-1 2^794-1 2^795-1 2^796-1 2^797-1 2^798-1 2^799-1 2^800-1 2^801-1 2^802-1 2^803-1 2^804-1 2^805-1 2^806-1 which looks like it's what you get by 1. listing the positive integers that aren't 2^k-1 2. following each n with the first n numbers 2^k-1 that haven't been taken yet with the proviso that 1 is deemed not to be of the form 2^k-1 :-). I expect this is either easy to prove or easy to refute, probably the former, but I have to go now. :-) -- g