A088012 has 28 Known Terms Now!
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Alex Violette
Feb 3, 2026, 2:55:52 PM (10 days ago) Feb 3
to seq...@googlegroups.com
Hello SeqFans,
I have come to announce that the number of known terms for A088012 has officially quadrupled in the span from 2020 to 2025 from its previous known term count of 7(This also means Hasler's comment in A088011 is well outdated now). Below are all new/"new" terms over 30 digits long that I have found over the years, first number is the date I found them/date found by, second number is sigma(n)-2n and the third is n itself:
<8/1/20 30 3278298202600507814120339275775985(t-)
1/24/23 30 3133639738039068908629117662878760945(t-)
11/22/24 90 1960326205542141554690232016958706407178195
9/8/23 -90 211140706455502585854234365249681393756160045
5/22/23 90 2244533631333227183087737092877226830703835955155
4/19/23 90 180992054688737770272043528208285834799024205252607955(t-)
6/2/24 90 297092104984437333118450402928700081576944203259285864447955(t-)
12/4/25 90 65550359577992789267032834923888913490732066955612413722114559955(t-)
9/5/25 -90 2226678033172347576369746234634423141670507666325690657440628242438225965(t)
11/30/20 -90 7644350909445166782402084283654804786139717999288520974394487224777343298764845(t)
7/14/25 -210 18289904961553750393651016244537845617690465894286576827250526982280934751530311297006960745 (Note: e^210 is "VERY" close to this number, closest I've seen an odd term hug the e^|(sigma(n)-2n)| bound)
1/23/23 210 796849239305504404034785815364401676407939709002626254730791658497098668638213959544062691245975
3/9/23 210 11106968019882351978977535281293712921823692984723047740966074565019330577896447573661303651696535
1/16/25 -150 6804143460215764754223920329422086030811988703725939577358800713117123658461242314068255637740585035(t) (Second known case where |sigma(a(n+1),1)-2*a(n+1)|<|sigma(a(n),1)-2*a(n)| for this sequence. upper bound on when this happens)
12/24/23 210 6767708683217596731186834074414504693909822888011369321990641170294784098470317343100089159897810206719895
8/19/25 -210 1054837894628358978223520300431612956011895654484562546279680036044231309013502811478933143043775039835556756251497377930776359175952336982016105(t)
Largest term's factorization is 3*5*7*11*397*22469*24659*142330932313*10623110733747829*98781698377377037938636404068817*27798831169309082431040945124155595488532674828613421635892639774215877 and it is notable as both its first three digits and last three digits are of the form |(sigma(n)-2n)|/2. Any term marked with a t is an odd term of A326138, t- for the abundant version of that sequence. Currently am writing up a paper on these and more with a planned Q1-Q2 preprint release. Also, here's two large odd solutions(one each) to the equations sigma(n)-2n=-102 and sigma(n)-2n=-730 that I have found: n=51059903414411024400435(-102) and n=3413348941881420604576829524586308719879823725(-730) respectively.
<8/1/20 30 3278298202600507814120339275775985(t-)
1/24/23 30 3133639738039068908629117662878760945(t-)
11/22/24 90 1960326205542141554690232016958706407178195
9/8/23 -90 211140706455502585854234365249681393756160045
5/22/23 90 2244533631333227183087737092877226830703835955155
4/19/23 90 180992054688737770272043528208285834799024205252607955(t-)
6/2/24 90 297092104984437333118450402928700081576944203259285864447955(t-)
12/4/25 90 65550359577992789267032834923888913490732066955612413722114559955(t-)
9/5/25 -90 2226678033172347576369746234634423141670507666325690657440628242438225965(t)
11/30/20 -90 7644350909445166782402084283654804786139717999288520974394487224777343298764845(t)
7/14/25 -210 18289904961553750393651016244537845617690465894286576827250526982280934751530311297006960745 (Note: e^210 is "VERY" close to this number, closest I've seen an odd term hug the e^|(sigma(n)-2n)| bound)
1/23/23 210 796849239305504404034785815364401676407939709002626254730791658497098668638213959544062691245975
3/9/23 210 11106968019882351978977535281293712921823692984723047740966074565019330577896447573661303651696535
1/16/25 -150 6804143460215764754223920329422086030811988703725939577358800713117123658461242314068255637740585035(t) (Second known case where |sigma(a(n+1),1)-2*a(n+1)|<|sigma(a(n),1)-2*a(n)| for this sequence. upper bound on when this happens)
12/24/23 210 6767708683217596731186834074414504693909822888011369321990641170294784098470317343100089159897810206719895
8/19/25 -210 1054837894628358978223520300431612956011895654484562546279680036044231309013502811478933143043775039835556756251497377930776359175952336982016105(t)
Largest term's factorization is 3*5*7*11*397*22469*24659*142330932313*10623110733747829*98781698377377037938636404068817*27798831169309082431040945124155595488532674828613421635892639774215877 and it is notable as both its first three digits and last three digits are of the form |(sigma(n)-2n)|/2. Any term marked with a t is an odd term of A326138, t- for the abundant version of that sequence. Currently am writing up a paper on these and more with a planned Q1-Q2 preprint release. Also, here's two large odd solutions(one each) to the equations sigma(n)-2n=-102 and sigma(n)-2n=-730 that I have found: n=51059903414411024400435(-102) and n=3413348941881420604576829524586308719879823725(-730) respectively.
Best,
Alex Violette
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