

A075114


Perfect powers n such that 2n + 1 is a perfect power; the value of y^b in the solution of the Diophantine equation x^a  2y^b = 1.


9



4, 121, 144, 4900, 166464, 5654884, 192099600, 6525731524, 221682772224, 7530688524100, 255821727047184, 8690408031080164, 295218051329678400, 10028723337177985444, 340681375412721826704
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OFFSET

1,1


COMMENTS

Note that the first ten numbers in this sequence are all squares. Except for 121, these squares are the y^2 in the Pell equation x^2  2y^2 = 1, whose solutions (x,y) are in sequences A001541 and A001542. The equation x^a  2y^b = 1 is very similar to Catalan's equation x^a  y^b = 1, which has only one solution. Bennett shows that the equation x^2  2y^b = 1 has no solutions for b>2. Hence all the terms in this sequence are squares and solutions other than the Pell solutions must satisfy x^a  2y^2 = 1 for a>2. The one known solution is 3^5  2*11^2 = 1. Are there any others?  T. D. Noe, Mar 29 2006


REFERENCES

Mohammad K. Azarian, Diophantine Pair, Problem B881, Fibonacci Quarterly, Vol. 37, No. 3, August 1999, pp. 277278. Solution appeared in Vol. 38, No. 2, May 2000, pp. 183184.


LINKS

Table of n, a(n) for n=1..15.
M. A. Bennett, Products of Consecutive Integers, Bull. London Math. Soc. 36 (2004), 683694


FORMULA

Empirical G.f.: x*(117*x^44091*x^3+3951*x^2+19*x4) / ((x1)*(x^234*x+1)).  Colin Barker, Dec 21 2012


MATHEMATICA

pp = Select[ Range[10^8], Apply[ GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 & ]; Select[pp, Apply[GCD, Last[ Transpose[ FactorInteger[2# + 1]]]] > 1 & ]
lim=10^14; lst={}; k=2; While[n=Floor[lim^(1/k)]; n>1, lst=Join[lst, Range[2, n]^k]; k++ ]; lst=Union[lst]; Intersection[lst, (lst1)/2] (*T. D. Noe, Mar 29 2006 *)


CROSSREFS

Cf. A001597.
Cf. A117547 (square root of terms).
Sequence in context: A239187 A062081 A053881 * A017186 A098839 A227525
Adjacent sequences: A075111 A075112 A075113 * A075115 A075116 A075117


KEYWORD

more,nonn


AUTHOR

Zak Seidov, Oct 11 2002


EXTENSIONS

Extended by Robert G. Wilson v, Oct 15 2002
More terms from T. D. Noe, Mar 29 2006
More terms from T. D. Noe, Nov 19 2006


STATUS

approved



