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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.

%I A261328 #23 Jan 08 2018 01:58:32
%S A261328 10,640,7290,8954,40960,52728,55566,70434,156250,405000,466560,536250,
%T A261328 573056,960089,997920,1176490,2037960,2621440,3374592,3556224,3748745,
%U A261328 4379424,4507776,5005000,5314410,6527466,6742450,7778106,8938800,10000000,10214145,12065355
%N 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.
%C A261328 See A261296 for the smaller of the pairs and for additional comments.
%D A261328 H. E. Dudeney, 536 Puzzles & Curious Problems, Charles Scribner's Sons, New York, 1967, pp 56, 268, #177
%H A261328 Chai Wah Wu, <a href="/A261328/b261328.txt">Table of n, a(n) for n = 1..302</a>
%H A261328 Gianlino, in reply to Smci, <a href="https://answers.yahoo.com/question/index?qid=20110722023859AAsGZxn">Solution method for "integers with the difference between their cubes is a square, and v.v."</a>, Yahoo! answers, 2011
%e A261328 (6, 10) is a pair since 10^3 - 6^3 = 784 = 28^2, 10^2 - 6^2 = 64 = 4^3.
%o A261328 (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
%o A261328 (Python)
%o A261328 # generate sequences A261328 and A261296
%o A261328 from __future__ import division
%o A261328 from sympy import divisors
%o A261328 from gmpy2 import is_square
%o A261328 alist = []
%o A261328 for i in range(1,10000):
%o A261328     c = i**3
%o A261328     for d in divisors(c, generator=True):
%o A261328         d2 = c//d
%o A261328         if d >= d2:
%o A261328             m, r = divmod(d+d2,2)
%o A261328             if not r:
%o A261328                 n = m-d2
%o A261328                 if n > 0 and (m,n) not in alist and is_square(c*m+d2*n**2):
%o A261328                     alist.append((m,n))
%o A261328 A261328_list, A261296_list = zip(*sorted(alist)) # _Chai Wah Wu_, Aug 25 2015
%Y A261328 Cf. A000290, A000578, A001014, A261296.
%K A261328 nonn
%O A261328 1,1
%A A261328 _Pieter Post_, Aug 15 2015
%E A261328 Added a(6) and more terms added by _Chai Wah Wu_, Aug 17 2015