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

1,1

Comments

See A261296 for the smaller of the pairs and for additional comments.

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

Example

(6, 10) is a pair since 10^3 - 6^3 = 784 = 28^2, 10^2 - 6^2 = 64 = 4^3.

Prog

(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
(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

Crossrefs

Cf. A000290, A000578, A001014, A261296.

Sequence in context: A179889 A209472 A132543 * A280897 A099024 A126680

Adjacent sequences: A261325 A261326 A261327 * A261329 A261330 A261331

Keyword

nonn

Author

Pieter Post, Aug 15 2015

Extensions

Added a(6) and more terms added by Chai Wah Wu, Aug 17 2015

Status

approved