Notes on the 8th term of A120716

N. J. A. Sloane, with additional comments from from 
Max Alekseyev, Maximilian Hasler, Simon Plouffe and
Robert G. Wilson v.

Jul 20 2007

We have 
 8 -> 2.4 -> 24
 -> 2.3.4.6.8.12 -> 2346812
 -> 2.4.13.26.52.45131.90262.180524.586703.1173406 
 -> 2413265245131902621805245867031173406 -> ?

The divisors of that last number aree

1, 2, 3, 6, 37, 74, 111, 113, 222, 226, 339, 678, 4181, 8362, 12543, 25086,
3192525397, 6385050794, 9577576191, 19155152382, 118123439689, 236246879378,
354370319067, 360755369861, 708740638134, 721510739722, 1082266109583,
2164532219166, 13347948684857, 26695897369714, 40043846054571, 80087692109142,
30132785470246166539693, 60265570940492333079386, 90398356410738499619079,
180796712821476999238158, 1114913062399108161968641, 2229826124798216323937282,
3344739187197324485905923, 3405004758137816818985309, 6689478374394648971811846,
6810009516275633637970618, 10215014274413450456955927,
20430028548826900913911854, 125985176051099222302456433,
251970352102198444604912866, 377955528153297666907369299,
755911056306595333814738598, 96199682896113474519861511083121,
192399365792226949039723022166242, 288599048688340423559584533249363,
577198097376680847119169066498726, 3559388267156198557234875910075477,
7118776534312397114469751820150954, 10678164801468595671704627730226431,
10870564167260822620744350752392673, 21356329602937191343409255460452862,
21741128334521645241488701504785346, 32611692501782467862233052257178019,
65223385003564935724466104514356038, 402210874188650436967540977838528901,
804421748377300873935081955677057802, 1206632622565951310902622933515586703,
2413265245131902621805245867031173406 

If we throw away the first and last terms and concatenate the rest,
we get the number:

23637741111132222263396784181836212543250863192525397638\
    5050794957757619119155152382118123439689236246879378\
    3543703190673607553698617087406381347215107397221082\
    2661095832164532219166133479486848572669589736971440\
    0438460545718008769210914230132785470246166539693602\
    6557094049233307938690398356410738499619079180796712\
    8214769992381581114913062399108161968641222982612479\
    8216323937282334473918719732448590592334050047581378\
    1681898530966894783743946489718118466810009516275633\
    6379706181021501427441345045695592720430028548826900\
    9139118541259851760510992223024564332519703521021984\
    4460491286637795552815329766690736929975591105630659\
    5333814738598961996828961134745198615110831211923993\
    6579222694903972302216624228859904868834042355958453\
    3249363577198097376680847119169066498726355938826715\
    6198557234875910075477711877653431239711446975182015\
    0954106781648014685956717046277302264311087056416726\
    0822620744350752392673213563296029371913434092554604\
    5286221741128334521645241488701504785346326116925017\
    8246786223305225717801965223385003564935724466104514\
    3560384022108741886504369675409778385289018044217483\
    7730087393508195567705780212066326225659513109026229\
    33515586703

which is the product of 13, 47 and a large composite number
with no factors below 10^8:
(But see the end of this file!)


Maximilian Hasler writes:
I confirm, according to PARI/gp, 13 and 47 divide,
the quotient is not prime but has no other divisors below 10^8.

(10:18) gp > big = eval( concat( {["",\\1,
2, 3, 6, 37, 74, 111, 113, 222, 226, 339, 678, 4181, 8362, 12543, 25086,
3192525397, 6385050794, 9577576191, 19155152382, 118123439689,
236246879378, 354370319067, 360755369861, 708740638134, 721510739722,
1082266109583, 2164532219166, 13347948684857, 26695897369714,
40043846054571, 80087692109142,
30132785470246166539693, 60265570940492333079386,
90398356410738499619079, 180796712821476999238158,
1114913062399108161968641, 2229826124798216323937282,
3344739187197324485905923, 3405004758137816818985309,
6689478374394648971811846,6810009516275633637970618,
10215014274413450456955927, 20430028548826900913911854,
125985176051099222302456433, 251970352102198444604912866,
377955528153297666907369299, 755911056306595333814738598,
96199682896113474519861511083121,
192399365792226949039723022166242, 288599048688340423559584533249363,
577198097376680847119169066498726, 3559388267156198557234875910075477,
7118776534312397114469751820150954, 10678164801468595671704627730226431,
10870564167260822620744350752392673, 21356329602937191343409255460452862,
21741128334521645241488701504785346, 32611692501782467862233052257178019,
65223385003564935724466104514356038, 402210874188650436967540977838528901,
804421748377300873935081955677057802, 1206632622565951310902622933515586703
\\, 2413265245131902621805245867031173406
]}))
%4 = 2363774111113222226339678418183621254325086319252539763850507949577576191191551523821181234396892362468793783543703190673607553698617087406381347215107397221082266109583216453221916613347948684857266958973697144004384605457180087692109142301327854702461665396936026557094049233307938690398356410738499619079180796712821476999238158111491306239910816196864122298261247982163239372823344739187197324485905923340500475813781681898530966894783743946489718118466810009516275633637970618102150142744134504569559272043002854882690091391185412598517605109922230245643325197035210219844460491286637795552815329766690736929975591105630659533381473859896199682896113474519861511083121192399365792226949039723022166242288599048688340423559584533249363577198097376680847119169066498726355938826715619855723487591007547771187765343123971144697518201509541067816480146859567170462773022643110870564167260822620744350752392673213563296029371913434092554604528622174112833452164524148870150478534632611692501782467862233052257178019652233850035649357244661045143560384022108741886504369675409778385289018044217483773008739350819556770578021206632622565951310902622933515586703

(10:22) gp > default(primelimit,10^8)
%13 = 100000000

(10:23) gp > factor(big, 10^8)
%15 =
[13 1]

[47 1]

[3868697399530641941636134890644224638829928509414958696973008100781630427482081053717154229782147892747616667010970852166297141896263645509625772856149586286550353698172203687760910987476184426934970472952041152216668748702422402114744913750127421771623020289584331517338869448949163159408111965202127036136138783490706181668147558283946491391016065788648318000427574438892372132280433288358751758569404129853519600349888186413307638125668975362503630308456317527849878287459853712114812017584508190784876058990184705208985419625508040259439615564520835099252578063887414435097316679683531580282840146917660780572791474804632830660198811554658264620124571971390935370021474946643806533922993518368285051133041896970030017059835653900571789815381501434829537019916639114118422158472529656065013891310977983258899779612314191726182517514745566395771653268182597660331870740823422046749864583997742625778645487190202231253730559596455143297225087761332426895993722625448232652174361100837467269693073424276681271977118907093044248198620695026260464092281541912834511463780156153483445644874049455783589212339363043464436297907048498563937727186420004801171173
1]

(10:20) gp > isprime( big/13/47 )
%9 = 0

----

From Max Alekseyev (maxale(AT)gmail.com), Jul 20 2007:

Here are two more prime divisors of that last number:
 1303590319 and 2029994921.
The cofactor is still composite.

Robert G. Wilson v also found these two prime divisors, Jul 22 2007

----

From Max Alekseyev (maxale(AT)gmail.com), Jul 25 2007:

Another factor is 2635016943923513073981336313737132619.

So the known prime factors of this number are now

13,
47,
1303590319,
2029994921,
2635016943923513073981336313737132619.

The cofactor is still composite and requires further factorization.