A063068 -id:A063068 - cp's OEIS frontend

A063066 Numbers expressible as (a^2-1)(b^2-1).

9, 24, 45, 64, 72, 105, 120, 144, 189, 192, 225, 240, 280, 297, 360, 384, 429, 504, 525, 576, 585, 640, 672, 720, 765, 792, 840, 864, 945, 960, 969, 1080, 1144, 1152, 1197, 1200, 1225, 1320, 1344, 1449, 1485, 1512, 1560, 1584, 1680, 1725, 1792, 1800, 1872

Offset: 1

Henry Bottomley, Jul 08 2001

nonn

			45 is on the list because 45=(2^2-1)(4^2-1)

K. S. Brown, Numbers Expressible as (a^2-1)(b^2-1)

Cf. A057535, A063067, A063068.

With[{nn=50},Take[(#[[1]]^2-1)(#[[2]]^2-1)&/@Tuples[Range[2,nn],2]//Union,nn]] (* Harvey P. Dale, May 29 2017 *)

A063067 Numbers expressible as (a^2-1)(b^2-1) in at least 2 distinct ways (b>=a>1).

Original entry on oeis.org

360, 504, 2304, 2520, 2880, 3465, 5040, 5400, 7920, 9360, 12285, 12600, 12672, 13440, 14400, 17325, 20160, 23040, 27720, 28224, 29400, 30600, 32760, 35640, 38080, 40320, 42840, 47880, 48384, 49920, 51480, 57960, 60480, 63360, 72072

Offset: 1

Henry Bottomley, Jul 08 2001

nonn

			360 is on the list since 360 = (2^2-1)(11^2-1) = (4^2-1)(5^2-1).

Donovan Johnson, Table of n, a(n) for n = 1..1000
K. S. Brown, Numbers Expressible as (a^2-1)(b^2-1)

Cf. A057535, A063066, A063068.

A057535 Numbers expressible as (a^2 - 1)*(b^2 - 1) in 5 distinct ways.

Original entry on oeis.org

588107520, 67270694400, 546939993600, 2128050512640, 37400697734400, 5566067918611200

Offset: 1

K. S. Brown (ksbrown(AT)seanet.com), Fred W. Helenius (fredh(AT)ix.netcom.com), Dean Hickerson, Randall L Rathbun

nonn

The next term (if it exists) is greater than 2^70.

K. S. Brown, Numbers Expressible as (a^2-1)(b^2-1)
K. S. Brown, MathPages Most Wanted List, Problem #14.

Cf. A063066, A063067, A063068.

Cf. A134856, A134857, A134858 (identical?).

{ f(a,b) = (a+1)*(a-1)*(b+1)*(b-1) } ans=vector(6,x,[0,0]); clear=ans; { g(a) = b=divisors(a*a-1); l=length(b); b=b+vector(l,x,a); for(x=1,l/2,c=4*a*b[x]*(a+1)*(a-1)*(b[x]+1)*(b[x]-1)*(a*b[x]-1)/((b[x]-a)*(b[x]-a));
d=floor(sqrt(sqrt(c))); count=1; for( y=2,d, if (c%(y*y-1)==0,e=ceil(sqrt(c/(y*y-1))); if (f(y,e)==c,ans[count]=[y,e]; count=count+1,),); ); if ( count>5,print("g:",a," ",c," ",ans); ans=clear,); ); } { find()= for(n=560,10001,print(n); g(n)); }
Store program as text file, load gp, \r textfilename and then run function find() to search for a 7th entry.

{ f(a,b) = (a+1)*(a-1)*(b+1)*(b-1) } ans=vector(6,x,[0,0])

A134857 Numbers that can be written as (a^2-1)(b^2-1) in four or more distinct ways.

Original entry on oeis.org

241920, 1048320, 10200960, 25724160, 37255680, 93139200, 123963840, 245044800, 588107520, 819786240, 1407893760, 1871251200, 3758169600, 5886558720, 8553283200, 10783342080, 13470367680, 19769460480, 30791819520, 40446806400

Offset: 1

Herman Beeksma, Nov 13 2007

Subsequence of A134856. Contains A134858 as a subsequence.

			241920 = (4^2-1)(127^2-1) = (7^2-1)(71^2-1) = (9^2-1)(55^2-1) = (17^2-1)(29^2-1).

MathPages, Numbers Expressible As (a^2-1)(b^2-1).

Cf. A134856, A134858.

Cf. A057535, A063066, A063067, A063068.

A134856 Numbers that can be written as (a^2-1)(b^2-1) in three or more distinct ways.

Original entry on oeis.org

2880, 27720, 40320, 49920, 63360, 98280, 241920, 282744, 491400, 547200, 604800, 950400, 970200, 1048320, 1370880, 1614600, 1774080, 2489760, 2608320, 2882880, 2923200, 3931200, 4817400, 6126120, 7338240, 7673400, 8426880, 10200960

Offset: 1

Herman Beeksma, Nov 13 2007

Contains A134857 and A134858 as subsequences.

			2880 = (2^2-1)(31^2-1) = (3^2-1)(19^2-1) = (5^2-1)(11^2-1).

MathPages, Numbers Expressible As (a^2-1)(b^2-1).

Cf. A134857, A134858, A057535, A063066, A063067, A063068.

A134858 Numbers that can be written as (a^2 - 1)(b^2 - 1) in five or more distinct ways.

Original entry on oeis.org

588107520, 67270694400, 546939993600, 2128050512640, 37400697734400, 5566067918611200

Offset: 1

Herman Beeksma, Nov 13 2007

Subsequence of A134856 and A134857.
Depending on the interpretation of A057535, this is either the same or a supersequence of A057535. [R. J. Mathar, Oct 16 2009]
The next term (if it exists) is greater than 2^70.

			588107520 = (13^2 - 1)(1871^2 - 1) = (17^2 - 1)(1429^2 - 1) = (55^2 - 1)(441^2 - 1) = (79^2 - 1)(307^2 - 1) = (129^2 - 1)(188^2 - 1).

MathPages, Numbers Expressible As (a^2-1)(b^2-1).

Cf. A057535, A134857, A134856, A063067, A063066, A063068.

cp's OEIS Frontend

A063066 Numbers expressible as (a^2-1)(b^2-1).

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A063067 Numbers expressible as (a^2-1)(b^2-1) in at least 2 distinct ways (b>=a>1).

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A057535 Numbers expressible as (a^2 - 1)*(b^2 - 1) in 5 distinct ways.

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PARI

A134857 Numbers that can be written as (a^2-1)(b^2-1) in four or more distinct ways.

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A134856 Numbers that can be written as (a^2-1)(b^2-1) in three or more distinct ways.

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A134858 Numbers that can be written as (a^2 - 1)(b^2 - 1) in five or more distinct ways.

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