A261328 Larger of pairs (m, n), such that the difference of their squares is a cube and the difference of their cubes is a square.
10, 640, 7290, 8954, 40960, 52728, 55566, 70434, 156250, 405000, 466560, 536250, 573056, 960089, 997920, 1176490, 2037960, 2621440, 3374592, 3556224, 3748745, 4379424, 4507776, 5005000, 5314410, 6527466, 6742450, 7778106, 8938800, 10000000, 10214145, 12065355
Offset: 1
Keywords
Examples
(6, 10) is a pair since 10^3 - 6^3 = 784 = 28^2, 10^2 - 6^2 = 64 = 4^3.
References
- H. E. Dudeney, 536 Puzzles & Curious Problems, Charles Scribner's Sons, New York, 1967, pp 56, 268, #177
Links
- Chai Wah Wu, Table of n, a(n) for n = 1..302
- Gianlino, in reply to Smci, Solution method for "integers with the difference between their cubes is a square, and v.v.", Yahoo! answers, 2011
Programs
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PARI
is(n)=forstep(k=n-1,1,-1,issquare(n^3-k^3)&&ispower(n^2-k^2,3)&&return(k)) \\ M. F. Hasler, Aug 17 2015
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Python
# generate sequences A261328 and A261296 from _future_ import division from sympy import divisors from gmpy2 import is_square alist = [] for i in range(1,10000): c = i**3 for d in divisors(c, generator=True): d2 = c//d if d >= d2: m, r = divmod(d+d2,2) if not r: n = m-d2 if n > 0 and (m,n) not in alist and is_square(c*m+d2*n**2): alist.append((m,n)) A261328_list, A261296_list = zip(*sorted(alist)) # Chai Wah Wu, Aug 25 2015
Extensions
Added a(6) and more terms added by Chai Wah Wu, Aug 17 2015
Comments