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
360 is on the list since 360 = (2^2-1)(11^2-1) = (4^2-1)(5^2-1).
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
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{ 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.
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{ f(a,b) = (a+1)*(a-1)*(b+1)*(b-1) } ans=vector(6,x,[0,0])
A063068
Smallest number expressible as (a^2-1)(b^2-1) in at least n distinct ways [with b>=a>1].
Original entry on oeis.org
9, 360, 2880, 241920, 588107520
Offset: 1
a(4)=241920 since 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)
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
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).
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
2880 = (2^2-1)(31^2-1) = (3^2-1)(19^2-1) = (5^2-1)(11^2-1).
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
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).
A372497
Positive integers of the form k^2 - 1 that are the product of two other distinct positive integers of the form k^2 - 1.
Original entry on oeis.org
24, 120, 360, 840, 960, 1680, 3024, 4224, 5040, 7920, 11880, 17160, 22800, 24024, 32760, 36480, 43680, 57120, 70224, 73440, 83520, 93024, 116280, 121800, 143640, 175560, 201600, 212520, 241080, 255024, 303600, 330624, 358800, 421200, 491400, 570024, 591360
Offset: 1
120 is a term since 120 = 15*8 = (4^2 - 1)*(3^2 - 1) and 120 = 11^2 - 1.
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Rest[Take[With[{k2=Range[500]^2-1},Select[Union[Times@@@Subsets[k2,{2}]],IntegerQ[Sqrt[#+1]]&]],50]] (* Harvey P. Dale, Apr 20 2025 *)
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isok1(k) = issquare(k+1);
isok2(k) = fordiv(k, d, if (isok1(d) && isok1(k/d), return(1)));
isok(k) = isok1(k) && isok2(k); \\ Michel Marcus, May 04 2024
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from math import isqrt
def is_perfect_square(n): return isqrt(abs(n))**2 == n
limit = 10**17
sequence_entries = set()
for a in range(2, isqrt(isqrt(limit))+1):
u = a**2 - 1
for b in range(a+1, isqrt(limit//u+1)+1):
v = b**2 - 1
if(is_perfect_square(u*v + 1)): sequence_entries.add(u*v)
sequence_entries = sorted(sequence_entries)
for i, j in enumerate(sequence_entries, 1):
print(i, j)
Showing 1-7 of 7 results.
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