Search result of sequences x1,x2,…,xn of distinct positive integers such that 1/n=1/x1^2+1/x2^2+...1/xk^2.
Related link:Find all sequences x1,x2,…,xn of distinct positive integers such that 1/2=1/x1^2+1/x2^2+...1/xn^2
2<=n<100

No solutions for 61,67,71,79,83,89,97 were found.

There are interesting identities below.


1/(bc)^2 + 1/(ca)^2 = 1/(ab)^2 where a^2+b^2=c^2.

1/(abd)^2 + 1/(bcd)^2 + 1/(cda)^2 = 1/(abc)^2 where a^2+b^2+c^2=d^2.

We can continue this process.

1/(abd)^3 + 1/(bcd)^3 + 1/(cda)^3 = 1/(abc)^3 where a^3+b^3+c^3=d^3.

We can continue this process.


1/2 = 1/2^2+ 1/3^2+ 1/4^2+ 1/5^2+ 1/8^2+ 1/10^2+ 1/15^2+ 1/20^2+ 1/24^2+ 1/30^2+ 1/40^2+ 1/60^2+ 1/120^2

1/2 = 1/2^2+ 1/3^2+ 1/4^2+ 1/6^2+ 1/7^2+ 1/8^2+ 1/12^2+ 1/21^2+ 1/24^2+ 1/28^2+ 1/56^2+ 1/168^2

1/2 = 1/2^2+ 1/3^2+ 1/4^2+ 1/6^2+ 1/8^2+ 1/11^2+ 1/12^2+ 1/13^2+ 1/17^2+ 1/22^2+ 1/24^2+ 1/33^2+ 1/34^2
    + 1/39^2+ 1/44^2+ 1/51^2+ 1/52^2+ 1/68^2+ 1/88^2+ 1/102^2+ 1/104^2+ 1/132^2+ 1/136^2+ 1/143^2+ 1/156^2
    + 1/187^2+ 1/204^2+ 1/221^2+ 1/264^2+ 1/286^2+ 1/312^2+ 1/374^2+ 1/408^2+ 1/429^2+ 1/442^2+ 1/561^2+ 1/572^2
    + 1/663^2+ 1/748^2+ 1/858^2+ 1/884^2+ 1/1122^2+ 1/1144^2+ 1/1326^2+ 1/1496^2+ 1/1716^2+ 1/1768^2+ 1/2244^2+ 1/2431^2
    + 1/2652^2+ 1/3432^2+ 1/4488^2+ 1/4862^2+ 1/5304^2+ 1/7293^2+ 1/9724^2+ 1/14586^2+ 1/19448^2+ 1/29172^2+ 1/58344^2

1/3 = 1/2^2+ 1/5^2+ 1/6^2+ 1/10^2+ 1/15^2+ 1/30^2

1/3 = 1/2^2+ 1/5^2+ 1/8^2+ 1/10^2+ 1/12^2+ 1/15^2+ 1/20^2+ 1/24^2+ 1/30^2+ 1/40^2+ 1/60^2+ 1/120^2

1/4=1/3^2 + 1/4^2 + 1/5^2 + 1/8^2 + 1/10^2 + 1/12^2 + 1/24^2 + 1/30^2 + 1/40^2 + 1/60^2 + 1/120

1/5 = 1/3^2+ 1/5^2+ 1/6^2+ 1/9^2+ 1/15^2+ 1/18^2+ 1/30^2+ 1/90^2

1/5 = 1/3^2+ 1/4^2+ 1/8^2+ 1/12^2+ 1/24^2+ 1/30^2+ 1/40^2+ 1/60^2+ 1/120^2

1/5 = 1/3^2+ 1/4^2+ 1/8^2+ 1/15^2+ 1/20^2+ 1/24^2+ 1/30^2+ 1/40^2+ 1/60^2+ 1/120^2

1/6 = 1/3^2+ 1/5^2+ 1/10^2+ 1/15^2+ 1/30^2

1/6 = 1/3^2+ 1/6^2+ 1/7^2+ 1/14^2+ 1/21^2

1/6 = 1/3^2+ 1/5^2+ 1/9^2+ 1/18^2+ 1/90^2

1/6 = 1/3^2+ 1/6^2+ 1/8^2+ 1/12^2+ 1/20^2+ 1/24^2+ 1/40^2+ 1/60^2+ 1/120^2

1/6 = 1/4^2+ 1/5^2+ 1/6^2+ 1/8^2+ 1/10^2+ 1/12^2+ 1/24^2+ 1/30^2+ 1/40^2+ 1/60^2+ 1/120^2

1/7 = 1/4^2+ 1/5^2+ 1/7^2+ 1/10^2+ 1/14^2+ 1/20^2+ 1/28^2+ 1/35^2+ 1/70^2+ 1/140^2

1/7 = 1/3^2+ 1/8^2+ 1/12^2+ 1/14^2+ 1/24^2+ 1/28^2+ 1/42^2+ 1/56^2+ 1/84^2+ 1/168^2

1/7 = 1/4^2+ 1/6^2+ 1/7^2+ 1/8^2+ 1/12^2+ 1/14^2+ 1/21^2+ 1/24^2+ 1/56^2+ 1/84^2+ 1/168^2

1/8 = 1/3^2+ 1/12^2+ 1/15^2+ 1/20^2

1/8 = 1/3^2+ 1/12^2+ 1/14^2+ 1/28^2+ 1/42^2

1/8 = 1/4^2+ 1/5^2+ 1/7^2+ 1/28^2+ 1/35^2

1/8 = 1/3^2+ 1/10^2+ 1/20^2+ 1/30^2+ 1/60^2

1/8 = 1/4^2+ 1/5^2+ 1/10^2+ 1/12^2+ 1/15^2+ 1/30^2

1/8 = 1/4^2+ 1/6^2+ 1/7^2+ 1/12^2+ 1/14^2+ 1/21^2

1/8 = 1/4^2+ 1/5^2+ 1/8^2+ 1/15^2+ 1/24^2+ 1/40^2+ 1/120^2

1/9=1/4^2 + 1/6^2 + 1/9^2 + 1/12^2 + 1/36^2 + 1/45^2 + 1/60^2

1/10=1/5^2 + 1/6^2 + 1/7^2 + 1/14^2 + 1/15^2 + 1/21^2

1/11=1/4^2 + 1/9^2 + 1/11^2 + 1/12^2 + 1/36^2 + 1/132^2 + 1/198^2

1/12=1/5^2 + 1/6^2 + 1/10^2 + 1/15^2 + 1/30^2

1/13=1/4^2 + 1/10^2 + 1/20^2 + 1/26^2 + 1/52^2 + 1/130^2 + 1/260^2

1/14=1/5^2 + 1/6^2 + 1/21^2 + 1/35^2 + 1/42^2

1/15=1/6^2 + 1/7^2 + 1/10^2 + 1/14^2 + 1/21^2 + 1/30^2

1/16=1/5^2 + 1/10^2 + 1/12^2 + 1/15^2 + 1/30^2

1/17=1/5^2 + 1/10^2 + 1/12^2 + 1/30^2 + 1/51^2 + 1/68^2 + 1/85^2 + 1/255^2 + 1/340^2 + 1/510^2 + 1/1020^2

1/18=1/6^2 + 1/7^2 + 1/14^2 + 1/21^2

1/19=1/6^2 + 1/8^2 + 1/12^2 + 1/24^2 + 1/57^2 + 1/76^2 + 1/152^2 + 1/228^2 + 1/456^2

1/20=1/5^2 + 1/12^2 + 1/20^2 + 1/60^2 + 1/75^2 + 1/100^2

1/21=1/5^2 + 1/15^2 + 1/21^2 + 1/35^2 + 1/105^2

1/22=1/5^2 + 1/15^2 + 1/33^2 + 1/110^2 + 1/330^2

1/23=1/6^2 + 1/9^2 + 1/18^2 + 1/69^2 + 1/138^2 + 1/414^2

1/24=1/6^2 + 1/12^2 + 1/15^2 + 1/20^2

1/25=1/6^2 + 1/10^2 + 1/30^2 + 1/35^2 + 1/70^2 + 1/105^2

1/26=1/6^2 + 1/10^2 + 1/39^2 + 1/195^2

1/27=1/7^2 + 1/9^2 + 1/18^2 + 1/30^2 + 1/126^2 + 1/210^2

1/28=1/6^2 + 1/14^2 + 1/21^2 + 1/42^2

1/29=1/8^2 + 1/10^2 + 1/12^2 + 1/24^2 + 1/87^2 + 1/232^2 + 1/290^2 + 1/348^2 + 1/580^2 + 1/696^2 + 1/870^2

1/30=1/7^2 + 1/10^2 + 1/21^2 + 1/42^2 + 1/105^2

1/31=1/8^2 + 1/10^2 + 1/16^2 + 1/20^2 + 1/120^2 + 1/124^2 + 1/155^2 + 1/240^2 + 1/248^2 + 1/310^2 + 1/496^2 + 1/620^2

1/32=1/8^2 + 1/9^2 + 1/18^2 + 1/72^2

1/33=1/6^2 + 1/22^2 + 1/66^2 + 1/77^2 + 1/154^2 + 1/231^2

1/34=1/6^2 + 1/30^2 + 1/51^2 + 1/85^2

1/35=1/9^2 + 1/10^2 + 1/14^2 + 1/30^2 + 1/315^2 + 1/630^2

1/36=1/7^2 + 1/15^2 + 1/21^2 + 1/42^2 + 1/105^2

1/37=1/7^2 + 1/14^2 + 1/37^2 + 1/42^2 + 1/74^2 + 1/222^2 + 1/259^2 + 1/777^2

1/38=1/7^2 + 1/14^2 + 1/38^2 + 1/95^2 + 1/665^2

1/39=1/10^2 + 1/12^2 + 1/13^2 + 1/20^2 + 1/60^2 + 1/780^2

1/40=1/7^2 + 1/20^2 + 1/28^2 + 1/35^2

1/41=1/10^2 + 1/12^2 + 1/15^2 + 1/20^2 + 1/60^2 + 1/82^2 + 1/123^2 + 1/410^2 + 1/615^2 + 1/2460^2

1/42=1/7^2 + 1/18^2 + 1/63^2 + 1/126^2

1/43=1/8^2 + 1/15^2 + 1/20^2 + 1/40^2 + 1/172^2 + 1/258^2 + 1/344^2 + 1/645^2 + 1/860^2 + 1/1720^2

1/44=1/7^2 + 1/22^2 + 1/66^2 + 1/231^2 + 1/462^2

1/45=1/7^2 + 1/25^2 + 1/75^2 + 1/175^2 + 1/525^2

1/46=1/10^2 + 1/12^2 + 1/20^2 + 1/23^2 + 1/69^2 + 1/92^2 + 1/138^2 + 1/230^2 + 1/460^2

1/47=1/10^2 + 1/12^2 + 1/20^2 + 1/30^2 + 1/40^2 + 1/120^2 + 1/235^2 + 1/470^2 + 1/564^2 + 1/1410^2 + 1/1880^2

1/48=1/7^2 + 1/60^2 + 1/84^2 + 1/420^2

1/49=1/12^2 + 1/14^2 + 1/15^2 + 1/20^2 + 1/28^2 + 1/84^2

1/50=1/10^2 + 1/12^2 + 1/20^2 + 1/60^2 + 1/75^2 + 1/100^2

1/51=1/10^2 + 1/12^2 + 1/20^2 + 1/85^2 + 1/204^2 + 1/1020^2

1/52=1/10^2 + 1/13^2 + 1/26^2 + 1/30^2 + 1/39^2 + 1/130^2 + 1/390^2

1/53=1/10^2 + 1/15^2 + 1/20^2 + 1/24^2 + 1/106^2 + 1/120^2 + 1/212^2 + 1/530^2 + 1/795^2 + 1/1060^2 + 1/1272^2 + 1/6360^2

1/54=1/9^2 + 1/18^2 + 1/22^2 + 1/33^2 + 1/99^2

1/55=1/10^2 + 1/15^2 + 1/22^2 + 1/30^2 + 1/55^2 + 1/66^2

1/56=1/12^2 + 1/14^2 + 1/15^2 + 1/28^2 + 1/105^2

1/57=1/12^2 + 1/15^2 + 1/19^2 + 1/20^2 + 1/38^2 + 1/76^2 + 1/228^2

1/58=1/10^2 + 1/15^2 + 1/20^2 + 1/87^2 + 1/116^2 + 1/145^2 + 1/174^2 + 1/435^2 + 1/580^2 + 1/870^2 + 1/1740^2

1/59=1/14^2 + 1/21^2 + 1/24^2 + 1/42^2 + 1/56^2 + 1/354^2 + 1/708^2 + 1/826^2 + 1/1239^2 + 1/1416^2 + 1/1652^2 + 1/2478^2 + 1/3304^2

1/60=1/9^2 + 1/18^2 + 1/35^2 + 1/70^2 + 1/90^2 + 1/105^2

1/62=1/12^2 + 1/14^2 + 1/21^2 + 1/28^2 + 1/62^2 + 1/84^2 + 1/93^2 + 1/217^2 + 1/1302^2 + 1/2604^2

1/63=1/12^2 + 1/15^2 + 1/20^2 + 1/28^2 + 1/42^2 + 1/84^2

1/64=1/12^2 + 1/15^2 + 1/20^2 + 1/24^2

1/65=1/12^2 + 1/15^2 + 1/20^2 + 1/26^2 + 1/260^2 + 1/780^2

1/66=1/11^2 + 1/15^2 + 1/30^2 + 1/33^2 + 1/55^2 + 1/110^2

1/68=1/10^2 + 1/17^2 + 1/30^2 + 1/102^2 + 1/170^2 + 1/510^2

1/69=1/9^2 + 1/23^2 +  1/63^2 + 1/483^2 + 1/1449^2

1/70=1/10^2 + 1/21^2 + 1/30^2 + 1/35^2 + 1/105^2

1/72=1/12^2 + 1/15^2 + 1/20^2

1/73=1/10^2 + 1/20^2 + 1/30^2 + 1/146^2 + 1/219^2 + 1/292^2 + 1/365^2 + 1/1460^2 + 1/4380^2

1/74=1/10^2 + 1/20^2 + 1/37^2 + 1/60^2 + 1/555^2 + 1/740^2 + 1/2220^2

1/75=1/12^2 + 1/14^2 + 1/35^2 + 1/60^2 + 1/84^2 + 1/140^2

1/76=1/9^2 + 1/38^2 + 1/114^2 + 1/171^2 + 1/342^2

1/77=1/77^2 + 1/15^2 + 1/11^2 + 1/105^2 + 1/231^2

1/78=1/13^2 + 1/15^2 + 1/30^2 + 1/39^2 + 1/45^2 + 1/90^2 + 1/117^2

1/80=1/10^2 + 1/21^2 + 1/84^2 + 1/105^2

1/81=1/12^2 + 1/18^2 + 1/28^2 + 1/36^2 + 1/63^2 + 1/252^2

1/82= 1/10^2 + 1/25^2 + 1/41^2 + 1/2050^2

1/84=1/14^2 + 1/15^2 + 1/21^2 + 1/105^2

1/85=1/12^2 + 1/17^2 + 1/34^2 + 1/60^2 + 1/68^2 + 1/1020^2

1/86=1/10^2 + 1/25^2 + 1/215^2 + 1/430^2 + 1/1075^2

1/87=1/10^2 + 1/30^2 + 1/58^2 + 1/145^2 + 1/174^2 + 1/435^2

1/88=1/11^2 + 1/20^2 + 1/44^2 + 1/110^2

1/90=1/10^2 + 1/38^2 + 1/57^2 + 1/95^2

1/91=1/12^2 + 1/26^2 + 1/28^2 + 1/39^2 + 1/52^2 + 1/84^2 + 1/91^2

1/92=1/12^2 + 1/18^2 + 1/46^2 + 1/69^2 + 1/92^2 + 1/207^2 + 1/276^2 + 1/828^2

1/93=1/62^2 + 1/372^2 + 1/31^2 + 1/12^2 + 1/20^2 + 1/1860^2

1/94=1/12^2 + 1/21^2 + 1/28^2 + 1/94^2 + 1/188^2 + 1/329^2 + 1/3948^2

1/95=1/10^2 + 1/57^2 + 1/95^2 + 1/114^2 + 1/190^2 + 1/570^2

1/96=1/15^2 + 1/20^2 + 1/24^2 + 1/30^2 + 1/40^2

1/98=1/14^2 + 1/15^2 + 1/42^2 + 1/105^2

1/99=1/12^2 + 1/20^2 + 1/44^2 + 1/110^2 + 1/165^2 + 1/220^2