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A075276 Generalized Markoff numbers: union of numbers a, b, c, d satisfying the Markoff(4) equation a^2 + b^2 + c^2 + d^2 = 4*a*b*c*d. 2
1, 3, 11, 41, 131, 153, 571, 1561, 1803, 2131, 5761, 7953, 17291, 18601, 25091, 29681, 79291, 110771, 221651, 253353, 295681, 349451, 413403, 817961, 1542841, 2282281, 2453891, 2641211, 3018753, 3252611, 3487001, 4114771, 4867203, 5757961, 11141771 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a,b,c,d is a solution to a^2 + b^2 + c^2 + d^2 = 4*a*b*c*d if and only if 2a, 2b, 2c, 2d is a solution to a^2 + b^2 + c^2 + d^2 = a*b*c*d. - Shanzhen Gao, Sep 18 2013

LINKS

Xianwen Wang, Table of n, a(n) for n = 1..5555

FORMULA

If (a, b, c, d) satisfies Markoff(4) equation, then so does (a, b, c, 4abc - d) and 4abc - d = (a^2 + b^2 + c^2)/d.

EXAMPLE

Some solutions to Markoff(4) equation a^2 + b^2 + c^2 + d^2 = 4abcd are (1,1,1,1), (1,1,1,3), (1,1,3,11), (1,3,11,131), (3,11,131,17291), (1,1,11,41), (1,11,41,1803), (1,1,41,153), (1,3,131,1561), (1,11,131,5761), (1,1,153,571), (1,1,571,2131), (1,1,2131,7953), (1,3,1561,18601).

MATHEMATICA

MAX=10^10; data = NestWhile[Select[Union[Sort/@Flatten[Table[{a, b, c, 4 a b c-d}/.MapThread[Rule, {{a, b, c, d}, #}]&/@Map[RotateLeft[ii, #]&, Range[4]], {ii, #}], 1]], Max[#] < MAX&]&, {{1, 1, 1, 1}, {1, 1, 1, 3}}, UnsameQ, 2]; Take[data//Flatten//Union, 30] (* Xianwen Wang, Feb 23 2013 *)

CROSSREFS

Cf. A002559.

Sequence in context: A329261 A073622 A181863 * A242233 A294504 A086972

Adjacent sequences:  A075273 A075274 A075275 * A075277 A075278 A075279

KEYWORD

easy,nonn

AUTHOR

Paul D. Hanna, Sep 12 2002

EXTENSIONS

Sequence corrected by Xianwen Wang, Feb 23 2013

STATUS

approved

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Last modified January 25 17:16 EST 2020. Contains 331245 sequences. (Running on oeis4.)