[seqfan] Re: Another Collatz-like algorithm

Sat Oct 19 11:31:09 CEST 2019

Hi,
Is this the correct trajectory for those numbers you mentioned?

173: ->173+190=363->363+378=741->741+780=1521->1521+1540=3061->3061+3081=6142=3071->3071+3081=6152=3076=1538=769->769+780=1549->1549+1596=3145->3145+3160=6305->6305+6328=12633->12633+12720=25353->25353+25425=50778=25389->25389+25425=50814=25407->25407+25425=50832=25416=12708=6354=3177->3177+3240=6417->6417+6441=12858=6429->6429+6441=12870=6435->6435+6441=12876=6438=3219->3219+3240=6459->6459+6555=13014=6507->6507+6555=13062=6531->6531+6555=13086=6543->6543+6555=13098=6549->6549+6555=13104=6552=3276=1638=819->819+820=1639->1639+1653=3292=1646=823->823+861=1684=842=421->421+435=856=428=214=107->107+120=227->227+231=458=229->229+231=460=230=115->115+120=235->235+253=488=244=122=61->61+66=127->127+136=263->263+276=539->539+561=1100=550=275->275+276=551->551+561=1112=556=278=139->139+153=292=146=73->73+78=151->151+153=304=152=76=38=19->19+21=40=20=10=5->5+6=11->11+15=26=13->13+15=28=14=7->7+10=17->17+21=38
100 steps

285: ->285+300=585->585+595=1180=590=295->295+300=595->595+630=1225->1225+1275=2500=1250=625->625+630=1255->1255+1275=2530=1265->1265+1275=2540=1270=635->635+666=1301->1301+1326=2627->2627+2628=5255->5255+5356=10611->10611+10731=21342=10671->10671+10731=21402=10701->10701+10731=21432=10716=5358=2679->2679+2701=5380=2690=1345->1345+1378=2723->2723+2775=5498=2749->2749+2775=5524=2762=1381->1381+1431=2812=1406=703->703+741=1444=722=361->361+378=739->739+741=1480=740=370=185->185+190=375->375+378=753->753+780=1533->1533+1540=3073->3073+3081=6154=3077->3077+3081=6158=3079->3079+3081=6160=3080=1540=770=385->385+406=791->791+820=1611->1611+1653=3264=1632=816=408=204=102=51->51+55=106=53->53+55=108=54=27->27+28=55->55+66=121->121+136=257->257+276=533->533+561=1094=547->547+561=1108=554=277->277+300=577->577+595=1172=586=293->293+300=593->593+595=1188=594=297->297+300=597->597+630=1227->1227+1275=2502=1251->1251+1275=2526=1263->1263+1275=2538=1269->1269+1275=2544=1272=636=318=159->159+171=330=165->165+171=336=168=84=42=21->21+28=49->49+55=104=52=26=13->13+15=28=14=7->7+10=17->17+21=38=19->19+21=40=20=10=5->5+6=11->11+15=26
128 steps

331: ->331+351=682=341->341+351=692=346=173->173+190=363->363+378=741->741+780=1521->1521+1540=3061->3061+3081=6142=3071->3071+3081=6152=3076=1538=769->769+780=1549->1549+1596=3145->3145+3160=6305->6305+6328=12633->12633+12720=25353->25353+25425=50778=25389->25389+25425=50814=25407->25407+25425=50832=25416=12708=6354=3177->3177+3240=6417->6417+6441=12858=6429->6429+6441=12870=6435->6435+6441=12876=6438=3219->3219+3240=6459->6459+6555=13014=6507->6507+6555=13062=6531->6531+6555=13086=6543->6543+6555=13098=6549->6549+6555=13104=6552=3276=1638=819->819+820=1639->1639+1653=3292=1646=823->823+861=1684=842=421->421+435=856=428=214=107->107+120=227->227+231=458=229->229+231=460=230=115->115+120=235->235+253=488=244=122=61->61+66=127->127+136=263->263+276=539->539+561=1100=550=275->275+276=551->551+561=1112=556=278=139->139+153=292=146=73->73+78=151->151+153=304=152=76=38=19->19+21=40=20=10=5->5+6=11->11+15=26=13->13+15=28=14=7->7+10=17->17+21=38
105 steps

These three cases did indeed find the value 5 in their eventual cycle.

If this matches what you are defining, then I found everything would
eventually cycle with a cycle that contains 5 or 4 (I didn't come
across anything that seemed to grow infinitely).
61909 was the first number to cycle and avoid 5:

61909: ->61909+62128=124037->124037+124251=248288=124144=62072=31036=15518=7759->7759+7875=15634=7817->7817+7875=15692=7846=3923->3923+4005=7928=3964=1982=991->991+1035=2026=1013->1013+1035=2048=1024=512=256=128=64=32=16=8=4=2=1->1+3=4

I checked up to 354999 and nothing grew unboundedly and nothing seemed
to cycle within large numbers ("large" meaning a cycle with all
numbers larger than 5).

JN

On Sat, Oct 19, 2019 at 12:13 AM Ali Sada via SeqFan
<seqfan at list.seqfan.eu> wrote:
>
>
> Hi Everyone,
>
>
>
> Please seethe algorithm below.
>
>
>
> 1. Pick aninteger n>0
>
> 2. If n iseven, divide by 2. If n is odd, find the least triangular number T greater thenn and add n+T.
>
> 3. Repeatstep 2. with either n/2 or n+T
>
>
>
> Other thanthe powers of 2, the numbers I tested take one of two routes:
>
>
> a.      They go into loops that have5 at their lowest point.Ex. 31+36=67; 67+78=145; 145+153=298; 298/2=149; 149+153=302;302/2=151; 151+153=304; 304/16=19; 19+21=40; 40/8=5.5+6=11; 11+15=26; 26/2=13; 13+15=28; 28/4=7; 7+10=17; 17+21=38; 38/2=19; 19+21=40; 40/8=5.
>
>
>
> b.      They go up to a point wherethe software doesn’t work anymore. Maybe they go up to infinity, but I cannotconfirm. Ex.173, 285, 331, etc.
>
>
>
> Can we provethat numbers in the second category actually go to infinity? And if so, would theybe an interesting sequence?
>
>
>
> Best,
>
>
>
> Ali
>
>
> --
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